Sandpiles¶

Functions and classes for mathematical sandpiles.

Version: 2.4

AUTHOR:

David Perkinson (June 4, 2015) Upgraded from version 2.3 to 2.4.

MAJOR CHANGES

Eliminated dependence on 4ti2, substituting the use of Polyhedron methods. Thus, no optional packages are necessary.
Fixed bug in Sandpile.__init__ so that now multigraphs are handled correctly.
Created sandpiles to handle examples of Sandpiles in analogy with graphs, simplicial_complexes, and polytopes. In the process, we implemented a much faster way of producing the sandpile grid graph.
Added support for open and closed sandpile Markov chains.
Added support for Weierstrass points.
Implemented the Cori-Le Borgne algorithm for computing ranks of divisors on complete graphs.

NEW METHODS

Sandpile: avalanche_polynomial, genus, group_gens, help, jacobian_representatives, markov_chain, picard_representatives, smith_form, stable_configs, stationary_density, tutte_polynomial.

SandpileConfig: burst_size, help.

SandpileDivisor: help, is_linearly_equivalent, is_q_reduced, is_weierstrass_pt, polytope, polytope_integer_pts, q_reduced, rank, simulate_threshold, stabilize, weierstrass_div, weierstrass_gap_seq, weierstrass_pts, weierstrass_rank_seq.

MINOR CHANGES

The sink argument to Sandpile.__init__ now defaults to the first vertex.
A SandpileConfig or SandpileDivisor may now be multiplied by an integer.
Sped up __add__ method for SandpileConfig and SandpileDivisor.
Enhanced string representation of a Sandpile (via _repr_ and the name methods).
Recurrents for complete graphs and cycle graphs are computed more quickly.
The stabilization code for SandpileConfig has been made more efficient.
Added optional probability distribution arguments to add_random methods.

Marshall Hampton (2010-1-10) modified for inclusion as a module within Sage library.
David Perkinson (2010-12-14) added show3d(), fixed bug in resolution(), replaced elementary_divisors() with invariant_factors(), added show() for SandpileConfig and SandpileDivisor.
David Perkinson (2010-9-18): removed is_undirected, added show(), added verbose arguments to several functions to display SandpileConfigs and divisors as lists of integers
David Perkinson (2010-12-19): created separate SandpileConfig, SandpileDivisor, and Sandpile classes
David Perkinson (2009-07-15): switched to using config_to_list instead of .values(), thus fixing a few bugs when not using integer labels for vertices.
David Perkinson (2009): many undocumented improvements
David Perkinson (2008-12-27): initial version

EXAMPLES:

For general help, enter Sandpile.help(), SandpileConfig.help(), and SandpileDivisor.help(). Miscellaneous examples appear below.

A weighted directed graph given as a Python dictionary:

Sage

sage: from sage.sandpiles import *
sage: g = {0: {},
....:      1: {0: 1, 2: 1, 3: 1},
....:      2: {1: 1, 3: 1, 4: 1},
....:      3: {1: 1, 2: 1, 4: 1},
....:      4: {2: 1, 3: 1}}

Python

>>> from sage.all import *
>>> from sage.sandpiles import *
>>> g = {Integer(0): {},
...      Integer(1): {Integer(0): Integer(1), Integer(2): Integer(1), Integer(3): Integer(1)},
...      Integer(2): {Integer(1): Integer(1), Integer(3): Integer(1), Integer(4): Integer(1)},
...      Integer(3): {Integer(1): Integer(1), Integer(2): Integer(1), Integer(4): Integer(1)},
...      Integer(4): {Integer(2): Integer(1), Integer(3): Integer(1)}}

The associated sandpile with 0 chosen as the sink:

Sage

sage: S = Sandpile(g,0)

[Python]

>>> from sage.all import *
>>> S = Sandpile(g,Integer(0))

Or just:

Sage

sage: S = Sandpile(g)

Python

>>> from sage.all import *
>>> S = Sandpile(g)

A picture of the graph:

Sage

sage: S.show()                              # long time

[Python]

>>> from sage.all import *
>>> S.show()                              # long time

The relevant Laplacian matrices:

Sage

sage: S.laplacian()
[ 0  0  0  0  0]
[-1  3 -1 -1  0]
[ 0 -1  3 -1 -1]
[ 0 -1 -1  3 -1]
[ 0  0 -1 -1  2]
sage: S.reduced_laplacian()
[ 3 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]

Python

>>> from sage.all import *
>>> S.laplacian()
[ 0  0  0  0  0]
[-1  3 -1 -1  0]
[ 0 -1  3 -1 -1]
[ 0 -1 -1  3 -1]
[ 0  0 -1 -1  2]
>>> S.reduced_laplacian()
[ 3 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]

The number of elements of the sandpile group for S:

Sage

sage: S.group_order()
8

[Python]

>>> from sage.all import *
>>> S.group_order()
8

The structure of the sandpile group:

Sage

sage: S.invariant_factors()
[1, 1, 1, 8]

Python

>>> from sage.all import *
>>> S.invariant_factors()
[1, 1, 1, 8]

The elements of the sandpile group for S:

Sage

sage: S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 2, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 0},
 {1: 2, 2: 2, 3: 0, 4: 1},
 {1: 2, 2: 0, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 1}]

[Python]

>>> from sage.all import *
>>> S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 2, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 0},
 {1: 2, 2: 2, 3: 0, 4: 1},
 {1: 2, 2: 0, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 1}]

The maximal stable element (2 grains of sand on vertices 1, 2, and 3, and 1 grain of sand on vertex 4:

Sage

sage: S.max_stable()
{1: 2, 2: 2, 3: 2, 4: 1}
sage: S.max_stable().values()
[2, 2, 2, 1]

Python

>>> from sage.all import *
>>> S.max_stable()
{1: 2, 2: 2, 3: 2, 4: 1}
>>> S.max_stable().values()
[2, 2, 2, 1]

The identity of the sandpile group for S:

Sage

sage: S.identity()
{1: 2, 2: 2, 3: 2, 4: 0}

[Python]

>>> from sage.all import *
>>> S.identity()
{1: 2, 2: 2, 3: 2, 4: 0}

An arbitrary sandpile configuration:

Sage

sage: c = SandpileConfig(S,[1,0,4,-3])
sage: c.equivalent_recurrent()
{1: 2, 2: 2, 3: 2, 4: 0}

Python

>>> from sage.all import *
>>> c = SandpileConfig(S,[Integer(1),Integer(0),Integer(4),-Integer(3)])
>>> c.equivalent_recurrent()
{1: 2, 2: 2, 3: 2, 4: 0}

Some group operations:

Sage

sage: m = S.max_stable()
sage: i = S.identity()
sage: m.values()
[2, 2, 2, 1]
sage: i.values()
[2, 2, 2, 0]
sage: m + i    # coordinate-wise sum
{1: 4, 2: 4, 3: 4, 4: 1}
sage: m - i
{1: 0, 2: 0, 3: 0, 4: 1}
sage: m & i  # add, then stabilize
{1: 2, 2: 2, 3: 2, 4: 1}
sage: e = m + m
sage: e
{1: 4, 2: 4, 3: 4, 4: 2}
sage: ~e   # stabilize
{1: 2, 2: 2, 3: 2, 4: 0}
sage: a = -m
sage: a & m
{1: 0, 2: 0, 3: 0, 4: 0}
sage: a * m   # add, then find the equivalent recurrent
{1: 2, 2: 2, 3: 2, 4: 0}
sage: a^3  # a*a*a
{1: 2, 2: 2, 3: 2, 4: 1}
sage: a^(-1) == m
True
sage: a < m  # every coordinate of a is < that of m
True

[Python]

>>> from sage.all import *
>>> m = S.max_stable()
>>> i = S.identity()
>>> m.values()
[2, 2, 2, 1]
>>> i.values()
[2, 2, 2, 0]
>>> m + i    # coordinate-wise sum
{1: 4, 2: 4, 3: 4, 4: 1}
>>> m - i
{1: 0, 2: 0, 3: 0, 4: 1}
>>> m & i  # add, then stabilize
{1: 2, 2: 2, 3: 2, 4: 1}
>>> e = m + m
>>> e
{1: 4, 2: 4, 3: 4, 4: 2}
>>> ~e   # stabilize
{1: 2, 2: 2, 3: 2, 4: 0}
>>> a = -m
>>> a & m
{1: 0, 2: 0, 3: 0, 4: 0}
>>> a * m   # add, then find the equivalent recurrent
{1: 2, 2: 2, 3: 2, 4: 0}
>>> a**Integer(3)  # a*a*a
{1: 2, 2: 2, 3: 2, 4: 1}
>>> a**(-Integer(1)) == m
True
>>> a < m  # every coordinate of a is < that of m
True

Firing an unstable vertex returns resulting configuration:

Sage

sage: c = S.max_stable() + S.identity()
sage: c.fire_vertex(1)
{1: 1, 2: 5, 3: 5, 4: 1}
sage: c
{1: 4, 2: 4, 3: 4, 4: 1}

Python

>>> from sage.all import *
>>> c = S.max_stable() + S.identity()
>>> c.fire_vertex(Integer(1))
{1: 1, 2: 5, 3: 5, 4: 1}
>>> c
{1: 4, 2: 4, 3: 4, 4: 1}

Fire all unstable vertices:

Sage

sage: c.unstable()
[1, 2, 3]
sage: c.fire_unstable()
{1: 3, 2: 3, 3: 3, 4: 3}

[Python]

>>> from sage.all import *
>>> c.unstable()
[1, 2, 3]
>>> c.fire_unstable()
{1: 3, 2: 3, 3: 3, 4: 3}

Stabilize c, returning the resulting configuration and the firing vector:

Sage

sage: c.stabilize(True)
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 6, 2: 8, 3: 8, 4: 8}]
sage: c
{1: 4, 2: 4, 3: 4, 4: 1}
sage: S.max_stable() & S.identity() == c.stabilize()
True

Python

>>> from sage.all import *
>>> c.stabilize(True)
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 6, 2: 8, 3: 8, 4: 8}]
>>> c
{1: 4, 2: 4, 3: 4, 4: 1}
>>> S.max_stable() & S.identity() == c.stabilize()
True

The number of superstable configurations of each degree:

Sage

sage: S.h_vector()
[1, 3, 4]
sage: S.postulation()
2

[Python]

>>> from sage.all import *
>>> S.h_vector()
[1, 3, 4]
>>> S.postulation()
2

the saturated homogeneous toppling ideal:

Sage

sage: S.ideal()
Ideal (x1 - x0, x3*x2 - x0^2, x4^2 - x0^2, x2^3 - x4*x3*x0,
       x4*x2^2 - x3^2*x0, x3^3 - x4*x2*x0, x4*x3^2 - x2^2*x0) of
 Multivariate Polynomial Ring in x4, x3, x2, x1, x0 over Rational Field

Python

>>> from sage.all import *
>>> S.ideal()
Ideal (x1 - x0, x3*x2 - x0^2, x4^2 - x0^2, x2^3 - x4*x3*x0,
       x4*x2^2 - x3^2*x0, x3^3 - x4*x2*x0, x4*x3^2 - x2^2*x0) of
 Multivariate Polynomial Ring in x4, x3, x2, x1, x0 over Rational Field

its minimal free resolution:

Sage

sage: S.resolution()
'R^1 <-- R^7 <-- R^15 <-- R^13 <-- R^4'

[Python]

>>> from sage.all import *
>>> S.resolution()
'R^1 <-- R^7 <-- R^15 <-- R^13 <-- R^4'

and its Betti numbers:

Sage

sage: S.betti()
           0     1     2     3     4
------------------------------------
    0:     1     1     -     -     -
    1:     -     2     2     -     -
    2:     -     4    13    13     4
------------------------------------
total:     1     7    15    13     4

Python

>>> from sage.all import *
>>> S.betti()
           0     1     2     3     4
------------------------------------
    0:     1     1     -     -     -
    1:     -     2     2     -     -
    2:     -     4    13    13     4
------------------------------------
total:     1     7    15    13     4

Some various ways of creating Sandpiles:

Sage

sage: S = sandpiles.Complete(4) # for more options enter ``sandpile.TAB``
sage: S = sandpiles.Wheel(6)

[Python]

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4)) # for more options enter ``sandpile.TAB``
>>> S = sandpiles.Wheel(Integer(6))

A multidigraph with loops (vertices 0, 1, 2; for example, there is a directed edge from vertex 2 to vertex 1 of weight 3, which can be thought of as three directed edges of the form (2,3). There is also a single loop at vertex 2 and an edge (2,0) of weight 2):

Sage

sage: S = Sandpile({0:[1,2], 1:[0,0,2], 2:[0,0,1,1,1,2], 3:[2]})

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0):[Integer(1),Integer(2)], Integer(1):[Integer(0),Integer(0),Integer(2)], Integer(2):[Integer(0),Integer(0),Integer(1),Integer(1),Integer(1),Integer(2)], Integer(3):[Integer(2)]})

Using the graph library (vertex 1 is specified as the sink; omitting this would make the sink vertex 0 by default):

Sage

sage: S = Sandpile(graphs.PetersenGraph(),1)

[Python]

>>> from sage.all import *
>>> S = Sandpile(graphs.PetersenGraph(),Integer(1))

Distribution of avalanche sizes:

Sage

sage: S = sandpiles.Grid(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(1000):
....:     m = m.add_random()
....:     m, f = m.stabilize(True)
....:     a.append(sum(f.values()))

sage: p = list_plot([[log(i + 1), log(a.count(i))]
....:                for i in [0..max(a)] if a.count(i)])
sage: p.axes_labels(['log(N)', 'log(D(N))'])
sage: t = text("Distribution of avalanche sizes", (2,2), rgbcolor=(1,0,0))
sage: show(p + t, axes_labels=['log(N)', 'log(D(N))'])      # long time

Python

>>> from sage.all import *
>>> S = sandpiles.Grid(Integer(10),Integer(10))
>>> m = S.max_stable()
>>> a = []
>>> for i in range(Integer(1000)):
...     m = m.add_random()
...     m, f = m.stabilize(True)
...     a.append(sum(f.values()))

>>> p = list_plot([[log(i + Integer(1)), log(a.count(i))]
...                for i in (ellipsis_range(Integer(0),Ellipsis,max(a))) if a.count(i)])
>>> p.axes_labels(['log(N)', 'log(D(N))'])
>>> t = text("Distribution of avalanche sizes", (Integer(2),Integer(2)), rgbcolor=(Integer(1),Integer(0),Integer(0)))
>>> show(p + t, axes_labels=['log(N)', 'log(D(N))'])      # long time

Working with sandpile divisors:

Sage

sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, [0,0,0,5])
sage: E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
sage: D.is_linearly_equivalent(E)
True
sage: D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, [0,0,0,5])
sage: E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
sage: D.is_linearly_equivalent(E)
True
sage: D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
sage: D.rank()
2

sage: sorted(D.effective_div(), key=str)
[{0: 0, 1: 0, 2: 0, 3: 5},
 {0: 0, 1: 0, 2: 4, 3: 1},
 {0: 0, 1: 4, 2: 0, 3: 1},
 {0: 1, 1: 1, 2: 1, 3: 2},
 {0: 4, 1: 0, 2: 0, 3: 1}]
sage: sorted(D.effective_div(False))
[[0, 0, 0, 5], [0, 0, 4, 1], [0, 4, 0, 1], [1, 1, 1, 2], [4, 0, 0, 1]]
sage: D.rank()
2
sage: D.rank(True)
(2, {0: 2, 1: 1, 2: 0, 3: 0})
sage: E = D.rank(True)[1]  # E proves the rank is not 3
sage: E.values()
[2, 1, 0, 0]
sage: E.deg()
3
sage: rank(D - E)
-1
sage: (D - E).effective_div()
[]

sage: D.weierstrass_pts()
(0, 1, 2, 3)
sage: D.weierstrass_rank_seq(0)
(2, 1, 0, 0, 0, -1)
sage: D.weierstrass_pts()
(0, 1, 2, 3)
sage: D.weierstrass_rank_seq(0)
(2, 1, 0, 0, 0, -1)

[Python]

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(S, [Integer(0),Integer(0),Integer(0),Integer(5)])
>>> E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
>>> D.is_linearly_equivalent(E)
True
>>> D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
>>> S = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(S, [Integer(0),Integer(0),Integer(0),Integer(5)])
>>> E = D.stabilize(); E
{0: 1, 1: 1, 2: 1, 3: 2}
>>> D.is_linearly_equivalent(E)
True
>>> D.q_reduced()
{0: 4, 1: 0, 2: 0, 3: 1}
>>> D.rank()
2

>>> sorted(D.effective_div(), key=str)
[{0: 0, 1: 0, 2: 0, 3: 5},
 {0: 0, 1: 0, 2: 4, 3: 1},
 {0: 0, 1: 4, 2: 0, 3: 1},
 {0: 1, 1: 1, 2: 1, 3: 2},
 {0: 4, 1: 0, 2: 0, 3: 1}]
>>> sorted(D.effective_div(False))
[[0, 0, 0, 5], [0, 0, 4, 1], [0, 4, 0, 1], [1, 1, 1, 2], [4, 0, 0, 1]]
>>> D.rank()
2
>>> D.rank(True)
(2, {0: 2, 1: 1, 2: 0, 3: 0})
>>> E = D.rank(True)[Integer(1)]  # E proves the rank is not 3
>>> E.values()
[2, 1, 0, 0]
>>> E.deg()
3
>>> rank(D - E)
-1
>>> (D - E).effective_div()
[]

>>> D.weierstrass_pts()
(0, 1, 2, 3)
>>> D.weierstrass_rank_seq(Integer(0))
(2, 1, 0, 0, 0, -1)
>>> D.weierstrass_pts()
(0, 1, 2, 3)
>>> D.weierstrass_rank_seq(Integer(0))
(2, 1, 0, 0, 0, -1)

class sage.sandpiles.sandpile.Sandpile(g, sink=None)[source]¶

Bases: DiGraph

Class for Dhar’s abelian sandpile model.

all_k_config(k)[source]¶

The constant configuration with all values set to \(k\).

INPUT:

k – integer

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.all_k_config(7)
{1: 7, 2: 7, 3: 7}

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.all_k_config(Integer(7))
{1: 7, 2: 7, 3: 7}

all_k_div(k)[source]¶

The divisor with all values set to \(k\).

INPUT:

k – integer

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.all_k_div(7)
{0: 7, 1: 7, 2: 7, 3: 7, 4: 7}

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.all_k_div(Integer(7))
{0: 7, 1: 7, 2: 7, 3: 7, 4: 7}

avalanche_polynomial(multivariable=True)[source]¶

The avalanche polynomial. See NOTE for details.

INPUT:

multivariable – boolean (default: True)

OUTPUT: polynomial

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: s.avalanche_polynomial()
9*x0*x1*x2 + 2*x0*x1 + 2*x0*x2 + 2*x1*x2 + 3*x0 + 3*x1 + 3*x2 + 24
sage: s.avalanche_polynomial(False)
9*x0^3 + 6*x0^2 + 9*x0 + 24

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> s.avalanche_polynomial()
9*x0*x1*x2 + 2*x0*x1 + 2*x0*x2 + 2*x1*x2 + 3*x0 + 3*x1 + 3*x2 + 24
>>> s.avalanche_polynomial(False)
9*x0^3 + 6*x0^2 + 9*x0 + 24

Note

For each nonsink vertex \(v\), let \(x_v\) be an indeterminate. If \((r,v)\) is a pair consisting of a recurrent \(r\) and nonsink vertex \(v\), then for each nonsink vertex \(w\), let \(n_w\) be the number of times vertex \(w\) fires in the stabilization of \(r + v\). Let \(M(r,v)\) be the monomial \(\prod_w x_w^{n_w}\), i.e., the exponent records the vector of \(n_w\) as \(w\) ranges over the nonsink vertices. The avalanche polynomial is then the sum of \(M(r,v)\) as \(r\) ranges over the recurrents and \(v\) ranges over the nonsink vertices. If multivariable is False, then set all the indeterminates equal to each other (and, thus, only count the number of vertex firings in the stabilizations, forgetting which particular vertices fired).

betti(verbose=True)[source]¶

Return the Betti table for the homogeneous toppling ideal.

If verbose is True, it prints the standard Betti table, otherwise, it returns a less formatted table.

INPUT:

verbose – boolean (default: True)

OUTPUT: Betti numbers for the sandpile

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     2     -     -
    2:     -     4     9     4
------------------------------
total:     1     6     9     4
sage: S.betti(False)
[1, 6, 9, 4]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     2     -     -
    2:     -     4     9     4
------------------------------
total:     1     6     9     4
>>> S.betti(False)
[1, 6, 9, 4]

betti_complexes()[source]¶

The support-complexes with non-trivial homology. (See NOTE.)

OUTPUT: list (of pairs [divisors, corresponding simplicial complex])

EXAMPLES:

Sage

sage: S = Sandpile({0:{},1:{0: 1, 2: 1, 3: 4},2:{3: 5},3:{1: 1, 2: 1}},0)
sage: p = S.betti_complexes()
sage: p[0]
[{0: -8, 1: 5, 2: 4, 3: 1},
 Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}]
sage: S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'
sage: S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     5     5     -
    2:     -     -     -     1
------------------------------
total:     1     5     5     1
sage: len(p)
11
sage: p[0][1].homology()
{0: Z, 1: 0}
sage: p[-1][1].homology()
{0: 0, 1: 0, 2: Z}

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0):{},Integer(1):{Integer(0): Integer(1), Integer(2): Integer(1), Integer(3): Integer(4)},Integer(2):{Integer(3): Integer(5)},Integer(3):{Integer(1): Integer(1), Integer(2): Integer(1)}},Integer(0))
>>> p = S.betti_complexes()
>>> p[Integer(0)]
[{0: -8, 1: 5, 2: 4, 3: 1},
 Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}]
>>> S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'
>>> S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     5     5     -
    2:     -     -     -     1
------------------------------
total:     1     5     5     1
>>> len(p)
11
>>> p[Integer(0)][Integer(1)].homology()
{0: Z, 1: 0}
>>> p[-Integer(1)][Integer(1)].homology()
{0: 0, 1: 0, 2: Z}

Note

A support-complex is the simplicial complex formed from the supports of the divisors in a linear system.

burning_config()[source]¶

The minimal burning configuration.

OUTPUT:

dict (configuration)

EXAMPLES:

Sage

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1},
....:      3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: S = Sandpile(g,0)
sage: S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
sage: S.burning_config().values()
[2, 0, 1, 1, 0]
sage: S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
sage: script = S.burning_script().values()
sage: script
[1, 3, 5, 1, 4]
sage: matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]

Python

>>> from sage.all import *
>>> g = {Integer(0):{},Integer(1):{Integer(0):Integer(1),Integer(3):Integer(1),Integer(4):Integer(1)},Integer(2):{Integer(0):Integer(1),Integer(3):Integer(1),Integer(5):Integer(1)},
...      Integer(3):{Integer(2):Integer(1),Integer(5):Integer(1)},Integer(4):{Integer(1):Integer(1),Integer(3):Integer(1)},Integer(5):{Integer(2):Integer(1),Integer(3):Integer(1)}}
>>> S = Sandpile(g,Integer(0))
>>> S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
>>> S.burning_config().values()
[2, 0, 1, 1, 0]
>>> S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
>>> script = S.burning_script().values()
>>> script
[1, 3, 5, 1, 4]
>>> matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]

Note

The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.

A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if \(b\) is the burning configuration, \(\sigma\) is its script, and \(\tilde{L}\) is the reduced Laplacian, then \(\sigma\cdot \tilde{L} = b\). The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).

The following are equivalent for a configuration \(c\) with burning configuration \(b\) having script \(\sigma\):

\(c\) is recurrent;

\(c+b\) stabilizes to \(c\);

the firing vector for the stabilization of \(c+b\) is \(\sigma\).

burning_script()[source]¶

A script for the minimal burning configuration.

OUTPUT: dictionary

EXAMPLES:

Sage

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1},
....:      3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: S = Sandpile(g,0)
sage: S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
sage: S.burning_config().values()
[2, 0, 1, 1, 0]
sage: S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
sage: script = S.burning_script().values()
sage: script
[1, 3, 5, 1, 4]
sage: matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]

Python

>>> from sage.all import *
>>> g = {Integer(0):{},Integer(1):{Integer(0):Integer(1),Integer(3):Integer(1),Integer(4):Integer(1)},Integer(2):{Integer(0):Integer(1),Integer(3):Integer(1),Integer(5):Integer(1)},
...      Integer(3):{Integer(2):Integer(1),Integer(5):Integer(1)},Integer(4):{Integer(1):Integer(1),Integer(3):Integer(1)},Integer(5):{Integer(2):Integer(1),Integer(3):Integer(1)}}
>>> S = Sandpile(g,Integer(0))
>>> S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
>>> S.burning_config().values()
[2, 0, 1, 1, 0]
>>> S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
>>> script = S.burning_script().values()
>>> script
[1, 3, 5, 1, 4]
>>> matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]

Note

The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.

A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if \(b\) is the burning configuration, \(s\) is its script, and \(L_{\mathrm{red}}\) is the reduced Laplacian, then \(s\cdot L_{\mathrm{red}}= b\). The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).

The following are equivalent for a configuration \(c\) with burning configuration \(b\) having script \(s\):

\(c\) is recurrent;

\(c+b\) stabilizes to \(c\);

the firing vector for the stabilization of \(c+b\) is \(s\).

canonical_divisor()[source]¶

The canonical divisor. This is the divisor with \(\deg(v)-2\) grains of sand on each vertex (not counting loops). Only for undirected graphs.

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.Complete(4)
sage: S.canonical_divisor()
{0: 1, 1: 1, 2: 1, 3: 1}
sage: s = Sandpile({0:[1,1],1:[0,0,1,1,1]},0)
sage: s.canonical_divisor()  # loops are disregarded
{0: 0, 1: 0}

Python

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4))
>>> S.canonical_divisor()
{0: 1, 1: 1, 2: 1, 3: 1}
>>> s = Sandpile({Integer(0):[Integer(1),Integer(1)],Integer(1):[Integer(0),Integer(0),Integer(1),Integer(1),Integer(1)]},Integer(0))
>>> s.canonical_divisor()  # loops are disregarded
{0: 0, 1: 0}

Warning

The underlying graph must be undirected.

dict()[source]¶

A dictionary of dictionaries representing a directed graph.

OUTPUT: dictionary

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.dict()
{0: {1: 1, 2: 1},
 1: {0: 1, 2: 1, 3: 1},
 2: {0: 1, 1: 1, 3: 1},
 3: {1: 1, 2: 1}}
sage: S.sink()
0

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.dict()
{0: {1: 1, 2: 1},
 1: {0: 1, 2: 1, 3: 1},
 2: {0: 1, 1: 1, 3: 1},
 3: {1: 1, 2: 1}}
>>> S.sink()
0

genus()[source]¶

The genus: (# non-loop edges) - (# vertices) + 1.

This is only defined for undirected graphs.

OUTPUT: integer

EXAMPLES:

Sage

sage: sandpiles.Complete(4).genus()
3
sage: sandpiles.Cycle(5).genus()
1

Python

>>> from sage.all import *
>>> sandpiles.Complete(Integer(4)).genus()
3
>>> sandpiles.Cycle(Integer(5)).genus()
1

groebner()[source]¶

Return a Groebner basis for the homogeneous toppling ideal.

It is computed with respect to the standard sandpile ordering (see ring).

OUTPUT: Groebner basis

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.groebner()
[x3*x2^2 - x1^2*x0, x2^3 - x3*x1*x0, x3*x1^2 - x2^2*x0,
 x1^3 - x3*x2*x0, x3^2 - x0^2, x2*x1 - x0^2]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.groebner()
[x3*x2^2 - x1^2*x0, x2^3 - x3*x1*x0, x3*x1^2 - x2^2*x0,
 x1^3 - x3*x2*x0, x3^2 - x0^2, x2*x1 - x0^2]

group_gens(verbose=True)[source]¶

A minimal list of generators for the sandpile group.

If verbose is False then the generators are represented as lists of integers.

INPUT:

verbose – boolean (default: True)

OUTPUT:

list of SandpileConfig (or of lists of integers if verbose is False)

EXAMPLES:

Sage

sage: s = sandpiles.Cycle(5)
sage: s.group_gens()
[{1: 0, 2: 1, 3: 1, 4: 1}]
sage: s.group_gens()[0].order()
5
sage: s = sandpiles.Complete(5)
sage: s.group_gens(False)
[[2, 3, 2, 2], [2, 2, 3, 2], [2, 2, 2, 3]]
sage: [i.order() for i in s.group_gens()]
[5, 5, 5]
sage: s.invariant_factors()
[1, 5, 5, 5]

Python

>>> from sage.all import *
>>> s = sandpiles.Cycle(Integer(5))
>>> s.group_gens()
[{1: 0, 2: 1, 3: 1, 4: 1}]
>>> s.group_gens()[Integer(0)].order()
5
>>> s = sandpiles.Complete(Integer(5))
>>> s.group_gens(False)
[[2, 3, 2, 2], [2, 2, 3, 2], [2, 2, 2, 3]]
>>> [i.order() for i in s.group_gens()]
[5, 5, 5]
>>> s.invariant_factors()
[1, 5, 5, 5]

group_order()[source]¶

The size of the sandpile group.

OUTPUT: integer

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.group_order()
11

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.group_order()
11

h_vector()[source]¶

The number of superstable configurations in each degree. Equivalently, this is the list of first differences of the Hilbert function of the (homogeneous) toppling ideal.

OUTPUT: list of nonnegative integers

EXAMPLES:

Sage

sage: s = sandpiles.Grid(2,2)
sage: s.hilbert_function()
[1, 5, 15, 35, 66, 106, 146, 178, 192]
sage: s.h_vector()
[1, 4, 10, 20, 31, 40, 40, 32, 14]

Python

>>> from sage.all import *
>>> s = sandpiles.Grid(Integer(2),Integer(2))
>>> s.hilbert_function()
[1, 5, 15, 35, 66, 106, 146, 178, 192]
>>> s.h_vector()
[1, 4, 10, 20, 31, 40, 40, 32, 14]

static help(verbose=True)[source]¶

List of Sandpile-specific methods (not inherited from Graph). If verbose, include short descriptions.

INPUT:

verbose – boolean (default: True)

OUTPUT: printed string

EXAMPLES:

Sage

sage: Sandpile.help() # long time
For detailed help with any method FOO listed below,
enter "Sandpile.FOO?" or enter "S.FOO?" for any Sandpile S.

all_k_config             -- The constant configuration with all values set to k.
all_k_div                -- The divisor with all values set to k.
avalanche_polynomial     -- The avalanche polynomial.
betti                    -- Return the Betti table for the homogeneous toppling ideal.
betti_complexes          -- The support-complexes with non-trivial homology.
burning_config           -- The minimal burning configuration.
burning_script           -- A script for the minimal burning configuration.
canonical_divisor        -- The canonical divisor.
dict                     -- A dictionary of dictionaries representing a directed graph.
genus                    -- The genus: (# non-loop edges) - (# vertices) + 1.
groebner                 -- Return a Groebner basis for the homogeneous toppling ideal.
group_gens               -- A minimal list of generators for the sandpile group.
group_order              -- The size of the sandpile group.
h_vector                 -- The number of superstable configurations in each degree.
help                     -- List of Sandpile-specific methods (not inherited from ...Graph...).
hilbert_function         -- The Hilbert function of the homogeneous toppling ideal.
ideal                    -- The saturated homogeneous toppling ideal.
identity                 -- The identity configuration.
in_degree                -- The in-degree of a vertex or a list of all in-degrees.
invariant_factors        -- The invariant factors of the sandpile group.
is_undirected            -- Is the underlying graph undirected?
jacobian_representatives -- Representatives for the elements of the Jacobian group.
laplacian                -- Return the Laplacian matrix of the graph.
markov_chain             -- The sandpile Markov chain for configurations or divisors.
max_stable               -- The maximal stable configuration.
max_stable_div           -- The maximal stable divisor.
max_superstables         -- The maximal superstable configurations.
min_recurrents           -- The minimal recurrent elements.
nonsink_vertices         -- The nonsink vertices.
nonspecial_divisors      -- The nonspecial divisors.
out_degree               -- The out-degree of a vertex or a list of all out-degrees.
picard_representatives   -- Representatives of the divisor classes of degree d in the Picard group.
points                   -- Generators for the multiplicative group of zeros of the sandpile ideal.
postulation              -- The postulation number of the toppling ideal.
recurrents               -- The recurrent configurations.
reduced_laplacian        -- Return the reduced Laplacian matrix of the graph.
reorder_vertices         -- A copy of the sandpile with vertex names permuted.
resolution               -- A minimal free resolution of the homogeneous toppling ideal.
ring                     -- The ring containing the homogeneous toppling ideal.
show                     -- Draw the underlying graph.
show3d                   -- Draw the underlying graph.
sink                     -- The sink vertex.
smith_form               -- The Smith normal form for the Laplacian.
solve                    -- Approximations of the complex affine zeros of the sandpile ideal.
stable_configs           -- Generator for all stable configurations.
stationary_density       -- The stationary density of the sandpile.
superstables             -- The superstable configurations.
symmetric_recurrents     -- The symmetric recurrent configurations.
tutte_polynomial         -- The Tutte polynomial of the underlying graph.
unsaturated_ideal        -- The unsaturated, homogeneous toppling ideal.
version                  -- The version number of Sage Sandpiles.
zero_config              -- The all-zero configuration.
zero_div                 -- The all-zero divisor.

Python

>>> from sage.all import *
>>> Sandpile.help() # long time
For detailed help with any method FOO listed below,
enter "Sandpile.FOO?" or enter "S.FOO?" for any Sandpile S.
<BLANKLINE>
all_k_config             -- The constant configuration with all values set to k.
all_k_div                -- The divisor with all values set to k.
avalanche_polynomial     -- The avalanche polynomial.
betti                    -- Return the Betti table for the homogeneous toppling ideal.
betti_complexes          -- The support-complexes with non-trivial homology.
burning_config           -- The minimal burning configuration.
burning_script           -- A script for the minimal burning configuration.
canonical_divisor        -- The canonical divisor.
dict                     -- A dictionary of dictionaries representing a directed graph.
genus                    -- The genus: (# non-loop edges) - (# vertices) + 1.
groebner                 -- Return a Groebner basis for the homogeneous toppling ideal.
group_gens               -- A minimal list of generators for the sandpile group.
group_order              -- The size of the sandpile group.
h_vector                 -- The number of superstable configurations in each degree.
help                     -- List of Sandpile-specific methods (not inherited from ...Graph...).
hilbert_function         -- The Hilbert function of the homogeneous toppling ideal.
ideal                    -- The saturated homogeneous toppling ideal.
identity                 -- The identity configuration.
in_degree                -- The in-degree of a vertex or a list of all in-degrees.
invariant_factors        -- The invariant factors of the sandpile group.
is_undirected            -- Is the underlying graph undirected?
jacobian_representatives -- Representatives for the elements of the Jacobian group.
laplacian                -- Return the Laplacian matrix of the graph.
markov_chain             -- The sandpile Markov chain for configurations or divisors.
max_stable               -- The maximal stable configuration.
max_stable_div           -- The maximal stable divisor.
max_superstables         -- The maximal superstable configurations.
min_recurrents           -- The minimal recurrent elements.
nonsink_vertices         -- The nonsink vertices.
nonspecial_divisors      -- The nonspecial divisors.
out_degree               -- The out-degree of a vertex or a list of all out-degrees.
picard_representatives   -- Representatives of the divisor classes of degree d in the Picard group.
points                   -- Generators for the multiplicative group of zeros of the sandpile ideal.
postulation              -- The postulation number of the toppling ideal.
recurrents               -- The recurrent configurations.
reduced_laplacian        -- Return the reduced Laplacian matrix of the graph.
reorder_vertices         -- A copy of the sandpile with vertex names permuted.
resolution               -- A minimal free resolution of the homogeneous toppling ideal.
ring                     -- The ring containing the homogeneous toppling ideal.
show                     -- Draw the underlying graph.
show3d                   -- Draw the underlying graph.
sink                     -- The sink vertex.
smith_form               -- The Smith normal form for the Laplacian.
solve                    -- Approximations of the complex affine zeros of the sandpile ideal.
stable_configs           -- Generator for all stable configurations.
stationary_density       -- The stationary density of the sandpile.
superstables             -- The superstable configurations.
symmetric_recurrents     -- The symmetric recurrent configurations.
tutte_polynomial         -- The Tutte polynomial of the underlying graph.
unsaturated_ideal        -- The unsaturated, homogeneous toppling ideal.
version                  -- The version number of Sage Sandpiles.
zero_config              -- The all-zero configuration.
zero_div                 -- The all-zero divisor.

hilbert_function()[source]¶

The Hilbert function of the homogeneous toppling ideal.

OUTPUT: list of nonnegative integers

EXAMPLES:

Sage

sage: s = sandpiles.Wheel(5)
sage: s.hilbert_function()
[1, 5, 15, 31, 45]
sage: s.h_vector()
[1, 4, 10, 16, 14]

Python

>>> from sage.all import *
>>> s = sandpiles.Wheel(Integer(5))
>>> s.hilbert_function()
[1, 5, 15, 31, 45]
>>> s.h_vector()
[1, 4, 10, 16, 14]

ideal(gens=False)[source]¶

The saturated homogeneous toppling ideal. If gens is True, the generators for the ideal are returned instead.

INPUT:

gens – boolean (default: False)

OUTPUT: ideal or, optionally, the generators of an ideal

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.ideal()
Ideal (x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
       x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0)
 of Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
sage: S.ideal(True)
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]
sage: S.ideal().gens()  # another way to get the generators
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.ideal()
Ideal (x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
       x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0)
 of Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
>>> S.ideal(True)
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]
>>> S.ideal().gens()  # another way to get the generators
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]

identity(verbose=True)[source]¶

The identity configuration.

If verbose is False, the configuration is converted to a list of integers.

INPUT:

verbose – boolean (default: True)

OUTPUT: SandpileConfig or a list of integers

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.identity()
{1: 2, 2: 2, 3: 0}
sage: s.identity(False)
[2, 2, 0]
sage: s.identity() & s.max_stable() == s.max_stable()
True

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.identity()
{1: 2, 2: 2, 3: 0}
>>> s.identity(False)
[2, 2, 0]
>>> s.identity() & s.max_stable() == s.max_stable()
True

in_degree(v=None)[source]¶

The in-degree of a vertex or a list of all in-degrees.

INPUT:

v – (optional) vertex name

OUTPUT: integer or dict

EXAMPLES:

Sage

sage: s = sandpiles.House()
sage: s.in_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
sage: s.in_degree(2)
3

Python

>>> from sage.all import *
>>> s = sandpiles.House()
>>> s.in_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
>>> s.in_degree(Integer(2))
3

invariant_factors()[source]¶

The invariant factors of the sandpile group.

OUTPUT: list of integers

EXAMPLES:

Sage

sage: s = sandpiles.Grid(2,2)
sage: s.invariant_factors()
[1, 1, 8, 24]

Python

>>> from sage.all import *
>>> s = sandpiles.Grid(Integer(2),Integer(2))
>>> s.invariant_factors()
[1, 1, 8, 24]

is_undirected()[source]¶

Is the underlying graph undirected? True if \((u,v)\) is and edge if and only if \((v,u)\) is an edge, each edge with the same weight.

OUTPUT: boolean

EXAMPLES:

Sage

sage: sandpiles.Complete(4).is_undirected()
True
sage: s = Sandpile({0:[1,2], 1:[0,2], 2:[0]}, 0)
sage: s.is_undirected()
False

Python

>>> from sage.all import *
>>> sandpiles.Complete(Integer(4)).is_undirected()
True
>>> s = Sandpile({Integer(0):[Integer(1),Integer(2)], Integer(1):[Integer(0),Integer(2)], Integer(2):[Integer(0)]}, Integer(0))
>>> s.is_undirected()
False

jacobian_representatives(verbose=True)[source]¶

Representatives for the elements of the Jacobian group. If verbose is False, then lists representing the divisors are returned.

INPUT:

verbose – boolean (default: True)

OUTPUT: list of SandpileDivisor (or of lists representing divisors)

EXAMPLES:

For an undirected graph, divisors of the form s - deg(s)*sink as s varies over the superstables forms a distinct set of representatives for the Jacobian group.:

Sage

sage: s = sandpiles.Complete(3)
sage: s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
sage: s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(3))
>>> s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
>>> s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]

If the graph is directed, the representatives described above may by equivalent modulo the rowspan of the Laplacian matrix:

Sage

sage: s = Sandpile({0: {1: 1, 2: 2}, 1: {0: 2, 2: 4}, 2: {0: 4, 1: 2}},0)
sage: s.group_order()
28
sage: s.jacobian_representatives()
[{0: -5, 1: 3, 2: 2}, {0: -4, 1: 3, 2: 1}]

[Python]

>>> from sage.all import *
>>> s = Sandpile({Integer(0): {Integer(1): Integer(1), Integer(2): Integer(2)}, Integer(1): {Integer(0): Integer(2), Integer(2): Integer(4)}, Integer(2): {Integer(0): Integer(4), Integer(1): Integer(2)}},Integer(0))
>>> s.group_order()
28
>>> s.jacobian_representatives()
[{0: -5, 1: 3, 2: 2}, {0: -4, 1: 3, 2: 1}]

Let \(\tau\) be the nonnegative generator of the kernel of the transpose of the Laplacian, and let \(\tau_s\) be its sink component, then the sandpile group is isomorphic to the direct sum of the cyclic group of order \(\tau_s\) and the Jacobian group. In the example above, we have:

Sage

sage: s.laplacian().left_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[14  5  8]

Python

>>> from sage.all import *
>>> s.laplacian().left_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[14  5  8]

Note

The Jacobian group is the set of all divisors of degree zero modulo the integer rowspan of the Laplacian matrix.

laplacian()[source]¶

Return the Laplacian matrix of the graph.

Its rows encode the vertex firing rules.

OUTPUT: matrix

EXAMPLES:

Sage

sage: G = sandpiles.Diamond()
sage: G.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]

Python

>>> from sage.all import *
>>> G = sandpiles.Diamond()
>>> G.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]

Warning

The function laplacian_matrix should be avoided. It returns the indegree version of the Laplacian.

markov_chain(state, distrib=None)[source]¶

The sandpile Markov chain for configurations or divisors. The chain starts at state. See NOTE for details.

INPUT:

state – SandpileConfig, SandpileDivisor, or list representing one of these
distrib – (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)

OUTPUT: generator for Markov chain (see NOTE)

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: m = s.markov_chain([0,0,0])
sage: next(m)          # random
{1: 0, 2: 0, 3: 0}
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 1, 0]
sage: next(m).values() # random
[0, 2, 0]
sage: next(m).values() # random
[0, 2, 1]
sage: next(m).values() # random
[1, 2, 1]
sage: next(m).values() # random
[2, 2, 1]
sage: m = s.markov_chain(s.zero_div(), [0.1,0.1,0.1,0.7])
sage: next(m).values() # random
[0, 0, 0, 1]
sage: next(m).values() # random
[0, 0, 1, 1]
sage: next(m).values() # random
[0, 0, 1, 2]
sage: next(m).values() # random
[1, 1, 2, 0]
sage: next(m).values() # random
[1, 1, 2, 1]
sage: next(m).values() # random
[1, 1, 2, 2]
sage: next(m).values() # random
[1, 1, 2, 3]
sage: next(m).values() # random
[1, 1, 2, 4]
sage: next(m).values() # random
[1, 1, 3, 4]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> m = s.markov_chain([Integer(0),Integer(0),Integer(0)])
>>> next(m)          # random
{1: 0, 2: 0, 3: 0}
>>> next(m).values() # random
[0, 0, 0]
>>> next(m).values() # random
[0, 0, 0]
>>> next(m).values() # random
[0, 0, 0]
>>> next(m).values() # random
[0, 1, 0]
>>> next(m).values() # random
[0, 2, 0]
>>> next(m).values() # random
[0, 2, 1]
>>> next(m).values() # random
[1, 2, 1]
>>> next(m).values() # random
[2, 2, 1]
>>> m = s.markov_chain(s.zero_div(), [RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.7')])
>>> next(m).values() # random
[0, 0, 0, 1]
>>> next(m).values() # random
[0, 0, 1, 1]
>>> next(m).values() # random
[0, 0, 1, 2]
>>> next(m).values() # random
[1, 1, 2, 0]
>>> next(m).values() # random
[1, 1, 2, 1]
>>> next(m).values() # random
[1, 1, 2, 2]
>>> next(m).values() # random
[1, 1, 2, 3]
>>> next(m).values() # random
[1, 1, 2, 4]
>>> next(m).values() # random
[1, 1, 3, 4]

Note

The closed sandpile Markov chain has state space consisting of the configurations on a sandpile. It transitions from a state by choosing a vertex at random (according to the probability distribution distrib), dropping a grain of sand at that vertex, and stabilizing. If the chosen vertex is the sink, the chain stays at the current state.

The open sandpile Markov chain has state space consisting of the recurrent elements, i.e., the state space is the sandpile group. It transitions from the configuration \(c\) by choosing a vertex \(v\) at random according to distrib. The next state is the stabilization of \(c+v\). If \(v\) is the sink vertex, then the stabilization of \(c+v\) is defined to be \(c\).

Note that in either case, if distrib is specified, its length is equal to the total number of vertices (including the sink).

REFERENCES:

[Lev2014]

max_stable()[source]¶

The maximal stable configuration.

OUTPUT: SandpileConfig (the maximal stable configuration)

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.max_stable()
{1: 1, 2: 2, 3: 2, 4: 1}

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.max_stable()
{1: 1, 2: 2, 3: 2, 4: 1}

max_stable_div()[source]¶

The maximal stable divisor.

OUTPUT:

SandpileDivisor (the maximal stable divisor)

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.max_stable_div()
{0: 1, 1: 2, 2: 2, 3: 1}
sage: s.out_degree()
{0: 2, 1: 3, 2: 3, 3: 2}

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.max_stable_div()
{0: 1, 1: 2, 2: 2, 3: 1}
>>> s.out_degree()
{0: 2, 1: 3, 2: 3, 3: 2}

max_superstables(verbose=True)[source]¶

The maximal superstable configurations. If the underlying graph is undirected, these are the superstables of highest degree. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – boolean (default: True)

OUTPUT: tuple of SandpileConfig

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.superstables(False)
[[0, 0, 0],
 [0, 0, 1],
 [1, 0, 1],
 [0, 2, 0],
 [2, 0, 0],
 [0, 1, 1],
 [1, 0, 0],
 [0, 1, 0]]
sage: s.max_superstables(False)
[[1, 0, 1], [0, 2, 0], [2, 0, 0], [0, 1, 1]]
sage: s.h_vector()
[1, 3, 4]

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.superstables(False)
[[0, 0, 0],
 [0, 0, 1],
 [1, 0, 1],
 [0, 2, 0],
 [2, 0, 0],
 [0, 1, 1],
 [1, 0, 0],
 [0, 1, 0]]
>>> s.max_superstables(False)
[[1, 0, 1], [0, 2, 0], [2, 0, 0], [0, 1, 1]]
>>> s.h_vector()
[1, 3, 4]

min_recurrents(verbose=True)[source]¶

The minimal recurrent elements. If the underlying graph is undirected, these are the recurrent elements of least degree. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – boolean (default: True)

OUTPUT: list of SandpileConfig

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.recurrents(False)
[[2, 2, 1],
 [2, 2, 0],
 [1, 2, 0],
 [2, 0, 1],
 [0, 2, 1],
 [2, 1, 0],
 [1, 2, 1],
 [2, 1, 1]]
sage: s.min_recurrents(False)
[[1, 2, 0], [2, 0, 1], [0, 2, 1], [2, 1, 0]]
sage: [i.deg() for i in s.recurrents()]
[5, 4, 3, 3, 3, 3, 4, 4]

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.recurrents(False)
[[2, 2, 1],
 [2, 2, 0],
 [1, 2, 0],
 [2, 0, 1],
 [0, 2, 1],
 [2, 1, 0],
 [1, 2, 1],
 [2, 1, 1]]
>>> s.min_recurrents(False)
[[1, 2, 0], [2, 0, 1], [0, 2, 1], [2, 1, 0]]
>>> [i.deg() for i in s.recurrents()]
[5, 4, 3, 3, 3, 3, 4, 4]

nonsink_vertices()[source]¶

The nonsink vertices.

OUTPUT: list of vertices

EXAMPLES:

Sage

sage: s = sandpiles.Grid(2,3)
sage: s.nonsink_vertices()
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)]

Python

>>> from sage.all import *
>>> s = sandpiles.Grid(Integer(2),Integer(3))
>>> s.nonsink_vertices()
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)]

nonspecial_divisors(verbose=True)[source]¶

The nonspecial divisors. Only for undirected graphs. (See NOTE.)

INPUT:

verbose – boolean (default: True)

OUTPUT: list (of divisors)

EXAMPLES:

Sage

sage: S = sandpiles.Complete(4)
sage: ns = S.nonspecial_divisors()
sage: D = ns[0]
sage: D.values()
[-1, 0, 1, 2]
sage: D.deg()
2
sage: [i.effective_div() for i in ns]
[[], [], [], [], [], []]

Python

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4))
>>> ns = S.nonspecial_divisors()
>>> D = ns[Integer(0)]
>>> D.values()
[-1, 0, 1, 2]
>>> D.deg()
2
>>> [i.effective_div() for i in ns]
[[], [], [], [], [], []]

Note

The “nonspecial divisors” are those divisors of degree \(g-1\) with empty linear system. The term is only defined for undirected graphs. Here, \(g = |E| - |V| + 1\) is the genus of the graph (not counting loops as part of \(|E|\)). If verbose is False, the divisors are converted to lists of integers.

Warning

The underlying graph must be undirected.

out_degree(v=None)[source]¶

The out-degree of a vertex or a list of all out-degrees.

INPUT:

v – (optional) vertex name

OUTPUT: integer or dict

EXAMPLES:

Sage

sage: s = sandpiles.House()
sage: s.out_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
sage: s.out_degree(2)
3

Python

>>> from sage.all import *
>>> s = sandpiles.House()
>>> s.out_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
>>> s.out_degree(Integer(2))
3

picard_representatives(d, verbose=True)[source]¶

Representatives of the divisor classes of degree \(d\) in the Picard group.

(Also see the documentation for jacobian_representatives.)

INPUT:

d – integer
verbose – boolean (default: True)

OUTPUT: slist of SandpileDivisors (or lists representing divisors)

EXAMPLES:

Sage

sage: s = sandpiles.Complete(3)
sage: s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
sage: s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]
sage: s.picard_representatives(3,False)
[[3, 0, 0], [2, 0, 1], [2, 1, 0]]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(3))
>>> s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
>>> s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]
>>> s.picard_representatives(Integer(3),False)
[[3, 0, 0], [2, 0, 1], [2, 1, 0]]

points()[source]¶

Generators for the multiplicative group of zeros of the sandpile ideal.

OUTPUT: list of complex numbers

EXAMPLES:

The sandpile group in this example is cyclic, and hence there is a single generator for the group of solutions.

Sage

sage: S = sandpiles.Complete(4)
sage: S.points()
[[-I, I, 1], [-I, 1, I]]

Python

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4))
>>> S.points()
[[-I, I, 1], [-I, 1, I]]

postulation()[source]¶

The postulation number of the toppling ideal. This is the largest weight of a superstable configuration of the graph.

OUTPUT: nonnegative integer

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: s.postulation()
3

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> s.postulation()
3

recurrents(verbose=True)[source]¶

The recurrent configurations. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – boolean (default: True)

OUTPUT: list of recurrent configurations

EXAMPLES:

Sage

sage: r = Sandpile(graphs.HouseXGraph(),0).recurrents()
sage: r[:3]
[{1: 2, 2: 3, 3: 3, 4: 1},
 {1: 1, 2: 3, 3: 3, 4: 0},
 {1: 1, 2: 3, 3: 3, 4: 1}]
sage: sandpiles.Complete(4).recurrents(False)
[[2, 2, 2],
 [2, 2, 1],
 [2, 1, 2],
 [1, 2, 2],
 [2, 2, 0],
 [2, 0, 2],
 [0, 2, 2],
 [2, 1, 1],
 [1, 2, 1],
 [1, 1, 2],
 [2, 1, 0],
 [2, 0, 1],
 [1, 2, 0],
 [1, 0, 2],
 [0, 2, 1],
 [0, 1, 2]]
sage: sandpiles.Cycle(4).recurrents(False)
[[1, 1, 1], [0, 1, 1], [1, 0, 1], [1, 1, 0]]

Python

>>> from sage.all import *
>>> r = Sandpile(graphs.HouseXGraph(),Integer(0)).recurrents()
>>> r[:Integer(3)]
[{1: 2, 2: 3, 3: 3, 4: 1},
 {1: 1, 2: 3, 3: 3, 4: 0},
 {1: 1, 2: 3, 3: 3, 4: 1}]
>>> sandpiles.Complete(Integer(4)).recurrents(False)
[[2, 2, 2],
 [2, 2, 1],
 [2, 1, 2],
 [1, 2, 2],
 [2, 2, 0],
 [2, 0, 2],
 [0, 2, 2],
 [2, 1, 1],
 [1, 2, 1],
 [1, 1, 2],
 [2, 1, 0],
 [2, 0, 1],
 [1, 2, 0],
 [1, 0, 2],
 [0, 2, 1],
 [0, 1, 2]]
>>> sandpiles.Cycle(Integer(4)).recurrents(False)
[[1, 1, 1], [0, 1, 1], [1, 0, 1], [1, 1, 0]]

reduced_laplacian()[source]¶

Return the reduced Laplacian matrix of the graph.

OUTPUT: matrix

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]
sage: S.reduced_laplacian()
[ 3 -1 -1]
[-1  3 -1]
[-1 -1  2]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]
>>> S.reduced_laplacian()
[ 3 -1 -1]
[-1  3 -1]
[-1 -1  2]

Note

This is the Laplacian matrix with the row and column indexed by the sink vertex removed.

reorder_vertices()[source]¶

A copy of the sandpile with vertex names permuted.

After reordering, vertex \(u\) comes before vertex \(v\) in the list of vertices if \(u\) is closer to the sink.

OUTPUT: Sandpile

EXAMPLES:

Sage

sage: S = Sandpile({0:[1], 2:[0,1], 1:[2]})
sage: S.dict()
{0: {1: 1}, 1: {2: 1}, 2: {0: 1, 1: 1}}
sage: T = S.reorder_vertices()

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0):[Integer(1)], Integer(2):[Integer(0),Integer(1)], Integer(1):[Integer(2)]})
>>> S.dict()
{0: {1: 1}, 1: {2: 1}, 2: {0: 1, 1: 1}}
>>> T = S.reorder_vertices()

The vertices 1 and 2 have been swapped:

Sage

sage: T.dict()
{0: {1: 1}, 1: {0: 1, 2: 1}, 2: {0: 1}}

[Python]

>>> from sage.all import *
>>> T.dict()
{0: {1: 1}, 1: {0: 1, 2: 1}, 2: {0: 1}}

resolution(verbose=False)[source]¶

A minimal free resolution of the homogeneous toppling ideal.

If verbose is True, then all of the mappings are returned. Otherwise, the resolution is summarized.

INPUT:

verbose – boolean (default: False)

OUTPUT: free resolution of the toppling ideal

EXAMPLES:

Sage

sage: S = Sandpile({0: {}, 1: {0: 1, 2: 1, 3: 4}, 2: {3: 5}, 3: {1: 1, 2: 1}},0)
sage: S.resolution()  # a Gorenstein sandpile graph
'R^1 <-- R^5 <-- R^5 <-- R^1'
sage: S.resolution(True)
[
[ x1^2 - x3*x0 x3*x1 - x2*x0  x3^2 - x2*x1  x2*x3 - x0^2  x2^2 - x1*x0],

[ x3  x2   0  x0   0]  [ x2^2 - x1*x0]
[-x1 -x3  x2   0 -x0]  [-x2*x3 + x0^2]
[ x0  x1   0  x2   0]  [-x3^2 + x2*x1]
[  0   0 -x1 -x3  x2]  [x3*x1 - x2*x0]
[  0   0  x0  x1 -x3], [ x1^2 - x3*x0]
]
sage: r = S.resolution(True)
sage: r[0]*r[1]
[0 0 0 0 0]
sage: r[1]*r[2]
[0]
[0]
[0]
[0]
[0]

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0): {}, Integer(1): {Integer(0): Integer(1), Integer(2): Integer(1), Integer(3): Integer(4)}, Integer(2): {Integer(3): Integer(5)}, Integer(3): {Integer(1): Integer(1), Integer(2): Integer(1)}},Integer(0))
>>> S.resolution()  # a Gorenstein sandpile graph
'R^1 <-- R^5 <-- R^5 <-- R^1'
>>> S.resolution(True)
[
[ x1^2 - x3*x0 x3*x1 - x2*x0  x3^2 - x2*x1  x2*x3 - x0^2  x2^2 - x1*x0],
<BLANKLINE>
[ x3  x2   0  x0   0]  [ x2^2 - x1*x0]
[-x1 -x3  x2   0 -x0]  [-x2*x3 + x0^2]
[ x0  x1   0  x2   0]  [-x3^2 + x2*x1]
[  0   0 -x1 -x3  x2]  [x3*x1 - x2*x0]
[  0   0  x0  x1 -x3], [ x1^2 - x3*x0]
]
>>> r = S.resolution(True)
>>> r[Integer(0)]*r[Integer(1)]
[0 0 0 0 0]
>>> r[Integer(1)]*r[Integer(2)]
[0]
[0]
[0]
[0]
[0]

ring()[source]¶

The ring containing the homogeneous toppling ideal.

OUTPUT: ring

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.ring()
Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
sage: S.ring().gens()
(x3, x2, x1, x0)

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.ring()
Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
>>> S.ring().gens()
(x3, x2, x1, x0)

Note

The indeterminate xi corresponds to the \(i\)-th vertex as listed my the method vertices. The term-ordering is degrevlex with indeterminates ordered according to their distance from the sink (larger indeterminates are further from the sink).

show(**kwds)[source]¶

Draw the underlying graph.

INPUT:

kwds – (optional) arguments passed to the show method for Graph or DiGraph

EXAMPLES:

Sage

sage: S = Sandpile({0:[], 1:[0,3,4], 2:[0,3,5], 3:[2,5], 4:[1,1], 5:[2,4]})
sage: S.show()
sage: S.show(graph_border=True, edge_labels=True)

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0):[], Integer(1):[Integer(0),Integer(3),Integer(4)], Integer(2):[Integer(0),Integer(3),Integer(5)], Integer(3):[Integer(2),Integer(5)], Integer(4):[Integer(1),Integer(1)], Integer(5):[Integer(2),Integer(4)]})
>>> S.show()
>>> S.show(graph_border=True, edge_labels=True)

show3d(**kwds)[source]¶

Draw the underlying graph.

INPUT:

kwds – (optional) arguments passed to the show method for Graph or DiGraph

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.show3d()                    # long time

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.show3d()                    # long time

sink()[source]¶

The sink vertex.

OUTPUT: sink vertex

EXAMPLES:

Sage

sage: G = sandpiles.House()
sage: G.sink()
0
sage: H = sandpiles.Grid(2,2)
sage: H.sink()
(0, 0)
sage: type(H.sink())
<... 'tuple'>

Python

>>> from sage.all import *
>>> G = sandpiles.House()
>>> G.sink()
0
>>> H = sandpiles.Grid(Integer(2),Integer(2))
>>> H.sink()
(0, 0)
>>> type(H.sink())
<... 'tuple'>

smith_form()[source]¶

The Smith normal form for the Laplacian. In detail: a list of integer matrices \(D, U, V\) such that \(ULV = D\) where \(L\) is the transpose of the Laplacian, \(D\) is diagonal, and \(U\) and \(V\) are invertible over the integers.

OUTPUT: list of integer matrices

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D,U,V = s.smith_form()
sage: D
[1 0 0 0]
[0 4 0 0]
[0 0 4 0]
[0 0 0 0]
sage: U*s.laplacian()*V == D  # Laplacian symmetric => transpose not necessary
True

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D,U,V = s.smith_form()
>>> D
[1 0 0 0]
[0 4 0 0]
[0 0 4 0]
[0 0 0 0]
>>> U*s.laplacian()*V == D  # Laplacian symmetric => transpose not necessary
True

solve()[source]¶

Approximations of the complex affine zeros of the sandpile ideal.

OUTPUT: list of complex numbers

EXAMPLES:

Sage

sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0)
sage: Z = S.solve(); Z
[[-0.707107000000000 + 0.707107000000000*I,
  0.707107000000000 - 0.707107000000000*I],
 [-0.707107000000000 - 0.707107000000000*I,
  0.707107000000000 + 0.707107000000000*I],
 [-I, -I],
 [I, I],
 [0.707107000000000 + 0.707107000000000*I,
  -0.707107000000000 - 0.707107000000000*I],
 [0.707107000000000 - 0.707107000000000*I,
  -0.707107000000000 + 0.707107000000000*I],
 [1, 1],
 [-1, -1]]
sage: len(Z)
8
sage: S.group_order()
8

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0): {}, Integer(1): {Integer(2): Integer(2)}, Integer(2): {Integer(0): Integer(4), Integer(1): Integer(1)}}, Integer(0))
>>> Z = S.solve(); Z
[[-0.707107000000000 + 0.707107000000000*I,
  0.707107000000000 - 0.707107000000000*I],
 [-0.707107000000000 - 0.707107000000000*I,
  0.707107000000000 + 0.707107000000000*I],
 [-I, -I],
 [I, I],
 [0.707107000000000 + 0.707107000000000*I,
  -0.707107000000000 - 0.707107000000000*I],
 [0.707107000000000 - 0.707107000000000*I,
  -0.707107000000000 + 0.707107000000000*I],
 [1, 1],
 [-1, -1]]
>>> len(Z)
8
>>> S.group_order()
8

Note

The solutions form a multiplicative group isomorphic to the sandpile group. Generators for this group are given exactly by points().

stable_configs(smax=None)[source]¶

Generator for all stable configurations.

If smax is provided, then the generator gives all stable configurations less than or equal to smax. If smax does not represent a stable configuration, then each component of smax is replaced by the corresponding component of the maximal stable configuration.

INPUT:

smax – (optional) SandpileConfig or list representing a SandpileConfig

OUTPUT: generator for all stable configurations

EXAMPLES:

Sage

sage: s = sandpiles.Complete(3)
sage: a = s.stable_configs()
sage: next(a)
{1: 0, 2: 0}
sage: [i.values() for i in a]
[[0, 1], [1, 0], [1, 1]]
sage: b = s.stable_configs([1,0])
sage: list(b)
[{1: 0, 2: 0}, {1: 1, 2: 0}]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(3))
>>> a = s.stable_configs()
>>> next(a)
{1: 0, 2: 0}
>>> [i.values() for i in a]
[[0, 1], [1, 0], [1, 1]]
>>> b = s.stable_configs([Integer(1),Integer(0)])
>>> list(b)
[{1: 0, 2: 0}, {1: 1, 2: 0}]

stationary_density()[source]¶

The stationary density of the sandpile.

OUTPUT: rational number

EXAMPLES:

Sage

sage: s = sandpiles.Complete(3)
sage: s.stationary_density()
10/9

sage: s = Sandpile(digraphs.DeBruijn(2,2),'00')
sage: s.stationary_density()
9/8

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(3))
>>> s.stationary_density()
10/9

>>> s = Sandpile(digraphs.DeBruijn(Integer(2),Integer(2)),'00')
>>> s.stationary_density()
9/8

Note

The stationary density of a sandpile is the sum \(\sum_c (\deg(c) + \deg(s))\) where \(\deg(s)\) is the degree of the sink and the sum is over all recurrent configurations.

REFERENCES:

[Lev2014]

superstables(verbose=True)[source]¶

The superstable configurations. If verbose is False, the configurations are converted to lists of integers. Superstables for undirected graphs are also known as G-parking functions.

INPUT:

verbose – boolean (default: True)

OUTPUT: list of SandpileConfig

EXAMPLES:

Sage

sage: sp = Sandpile(graphs.HouseXGraph(),0).superstables()
sage: sp[:3]
[{1: 0, 2: 0, 3: 0, 4: 0},
 {1: 1, 2: 0, 3: 0, 4: 1},
 {1: 1, 2: 0, 3: 0, 4: 0}]
sage: sandpiles.Complete(4).superstables(False)
[[0, 0, 0],
 [0, 0, 1],
 [0, 1, 0],
 [1, 0, 0],
 [0, 0, 2],
 [0, 2, 0],
 [2, 0, 0],
 [0, 1, 1],
 [1, 0, 1],
 [1, 1, 0],
 [0, 1, 2],
 [0, 2, 1],
 [1, 0, 2],
 [1, 2, 0],
 [2, 0, 1],
 [2, 1, 0]]
sage: sandpiles.Cycle(4).superstables(False)
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]]

Python

>>> from sage.all import *
>>> sp = Sandpile(graphs.HouseXGraph(),Integer(0)).superstables()
>>> sp[:Integer(3)]
[{1: 0, 2: 0, 3: 0, 4: 0},
 {1: 1, 2: 0, 3: 0, 4: 1},
 {1: 1, 2: 0, 3: 0, 4: 0}]
>>> sandpiles.Complete(Integer(4)).superstables(False)
[[0, 0, 0],
 [0, 0, 1],
 [0, 1, 0],
 [1, 0, 0],
 [0, 0, 2],
 [0, 2, 0],
 [2, 0, 0],
 [0, 1, 1],
 [1, 0, 1],
 [1, 1, 0],
 [0, 1, 2],
 [0, 2, 1],
 [1, 0, 2],
 [1, 2, 0],
 [2, 0, 1],
 [2, 1, 0]]
>>> sandpiles.Cycle(Integer(4)).superstables(False)
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]]

symmetric_recurrents(orbits)[source]¶

The symmetric recurrent configurations.

INPUT:

orbits – list of lists partitioning the vertices

OUTPUT: list of recurrent configurations

EXAMPLES:

Sage

sage: S = Sandpile({0: {},
....:              1: {0: 1, 2: 1, 3: 1},
....:              2: {1: 1, 3: 1, 4: 1},
....:              3: {1: 1, 2: 1, 4: 1},
....:              4: {2: 1, 3: 1}})
sage: S.symmetric_recurrents([[1],[2,3],[4]])
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 2, 4: 0}]
sage: S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 2, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 0},
 {1: 2, 2: 2, 3: 0, 4: 1},
 {1: 2, 2: 0, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 1}]

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0): {},
...              Integer(1): {Integer(0): Integer(1), Integer(2): Integer(1), Integer(3): Integer(1)},
...              Integer(2): {Integer(1): Integer(1), Integer(3): Integer(1), Integer(4): Integer(1)},
...              Integer(3): {Integer(1): Integer(1), Integer(2): Integer(1), Integer(4): Integer(1)},
...              Integer(4): {Integer(2): Integer(1), Integer(3): Integer(1)}})
>>> S.symmetric_recurrents([[Integer(1)],[Integer(2),Integer(3)],[Integer(4)]])
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 2, 4: 0}]
>>> S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 2, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 0},
 {1: 2, 2: 2, 3: 0, 4: 1},
 {1: 2, 2: 0, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 0},
 {1: 2, 2: 1, 3: 2, 4: 1},
 {1: 2, 2: 2, 3: 1, 4: 1}]

Note

The user is responsible for ensuring that the list of orbits comes from a group of symmetries of the underlying graph.

tutte_polynomial()[source]¶

The Tutte polynomial of the underlying graph. Only defined for undirected sandpile graphs.

OUTPUT: polynomial

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: s.tutte_polynomial()
x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y
sage: s.tutte_polynomial().subs(x=1)
y^3 + 3*y^2 + 6*y + 6
sage: s.tutte_polynomial().subs(x=1).coefficients() == s.h_vector()
True

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> s.tutte_polynomial()
x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y
>>> s.tutte_polynomial().subs(x=Integer(1))
y^3 + 3*y^2 + 6*y + 6
>>> s.tutte_polynomial().subs(x=Integer(1)).coefficients() == s.h_vector()
True

unsaturated_ideal()[source]¶

The unsaturated, homogeneous toppling ideal.

OUTPUT: ideal

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.unsaturated_ideal().gens()
[x1^3 - x3*x2*x0, x2^3 - x3*x1*x0, x3^2 - x2*x1]
sage: S.ideal().gens()
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.unsaturated_ideal().gens()
[x1^3 - x3*x2*x0, x2^3 - x3*x1*x0, x3^2 - x2*x1]
>>> S.ideal().gens()
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0,
 x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]

static version()[source]¶

The version number of Sage Sandpiles.

OUTPUT: string

EXAMPLES:

Sage

sage: Sandpile.version()
Sage Sandpiles Version 2.4
sage: S = sandpiles.Complete(3)
sage: S.version()
Sage Sandpiles Version 2.4

Python

>>> from sage.all import *
>>> Sandpile.version()
Sage Sandpiles Version 2.4
>>> S = sandpiles.Complete(Integer(3))
>>> S.version()
Sage Sandpiles Version 2.4

zero_config()[source]¶

The all-zero configuration.

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: s.zero_config()
{1: 0, 2: 0, 3: 0}

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> s.zero_config()
{1: 0, 2: 0, 3: 0}

zero_div()[source]¶

The all-zero divisor.

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.zero_div()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0}

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.zero_div()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0}

class sage.sandpiles.sandpile.SandpileConfig(S, c)[source]¶

Bases: dict

Class for configurations on a sandpile.

add_random(distrib=None)[source]¶

Add one grain of sand to a random vertex.

Optionally, a probability distribution, distrib, may be placed on the vertices or the nonsink vertices.

See NOTE for details.

INPUT:

distrib – (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: c = s.zero_config()
sage: c.add_random() # random
{1: 0, 2: 1, 3: 0}
sage: c
{1: 0, 2: 0, 3: 0}
sage: c.add_random([0.1,0.1,0.8]) # random
{1: 0, 2: 0, 3: 1}
sage: c.add_random([0.7,0.1,0.1,0.1]) # random
{1: 0, 2: 0, 3: 0}

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> c = s.zero_config()
>>> c.add_random() # random
{1: 0, 2: 1, 3: 0}
>>> c
{1: 0, 2: 0, 3: 0}
>>> c.add_random([RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.8')]) # random
{1: 0, 2: 0, 3: 1}
>>> c.add_random([RealNumber('0.7'),RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.1')]) # random
{1: 0, 2: 0, 3: 0}

We compute the “sizes” of the avalanches caused by adding random grains of sand to the maximal stable configuration on a grid graph. The function stabilize() returns the firing vector of the stabilization, a dictionary whose values say how many times each vertex fires in the stabilization.:

Sage

sage: S = sandpiles.Grid(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(1000):
....:     m = m.add_random()
....:     m, f = m.stabilize(True)
....:     a.append(sum(f.values()))

sage: p = list_plot([[log(i + 1), log(a.count(i))]
....:                for i in [0..max(a)] if a.count(i)])
sage: p.axes_labels(['log(N)', 'log(D(N))'])
sage: t = text("Distribution of avalanche sizes", (2,2), rgbcolor=(1,0,0))
sage: show(p + t, axes_labels=['log(N)', 'log(D(N))'])      # long time

[Python]

>>> from sage.all import *
>>> S = sandpiles.Grid(Integer(10),Integer(10))
>>> m = S.max_stable()
>>> a = []
>>> for i in range(Integer(1000)):
...     m = m.add_random()
...     m, f = m.stabilize(True)
...     a.append(sum(f.values()))

>>> p = list_plot([[log(i + Integer(1)), log(a.count(i))]
...                for i in (ellipsis_range(Integer(0),Ellipsis,max(a))) if a.count(i)])
>>> p.axes_labels(['log(N)', 'log(D(N))'])
>>> t = text("Distribution of avalanche sizes", (Integer(2),Integer(2)), rgbcolor=(Integer(1),Integer(0),Integer(0)))
>>> show(p + t, axes_labels=['log(N)', 'log(D(N))'])      # long time

Note

If distrib is None, then the probability is the uniform probability on the nonsink vertices. Otherwise, there are two possibilities:

(i) the length of distrib is equal to the number of vertices, and distrib represents a probability distribution on all of the vertices. In that case, the sink may be chosen at random, in which case, the configuration is unchanged.

(ii) Otherwise, the length of distrib must be equal to the number of nonsink vertices, and distrib represents a probability distribution on the nonsink vertices.

Warning

If distrib != None, the user is responsible for assuring the sum of its entries is 1 and that its length is equal to the number of sink vertices or the number of nonsink vertices.

burst_size(v)[source]¶

Return the burst size of the configuration with respect to the given vertex.

INPUT:

v – vertex

OUTPUT: integer

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: [i.burst_size(0) for i in s.recurrents()]
[1, 1, 1, 1, 1, 1, 1, 1]
sage: [i.burst_size(1) for i in s.recurrents()]
[0, 0, 1, 2, 1, 2, 0, 2]

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> [i.burst_size(Integer(0)) for i in s.recurrents()]
[1, 1, 1, 1, 1, 1, 1, 1]
>>> [i.burst_size(Integer(1)) for i in s.recurrents()]
[0, 0, 1, 2, 1, 2, 0, 2]

Note

To define c.burst(v), if \(v\) is not the sink, let \(c'\) be the unique recurrent for which the stabilization of \(c' + v\) is \(c\). The burst size is then the amount of sand that goes into the sink during this stabilization. If \(v\) is the sink, the burst size is defined to be 1.

REFERENCES:

[Lev2014]

deg()[source]¶

The degree of the configuration.

OUTPUT: integer

EXAMPLES:

Sage

sage: S = sandpiles.Complete(3)
sage: c = SandpileConfig(S, [1,2])
sage: c.deg()
3

Python

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(3))
>>> c = SandpileConfig(S, [Integer(1),Integer(2)])
>>> c.deg()
3

dualize()[source]¶

The difference with the maximal stable configuration.

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: S.max_stable()
{1: 1, 2: 1}
sage: c.dualize()
{1: 0, 2: -1}
sage: S.max_stable() - c == c.dualize()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> c = SandpileConfig(S, [Integer(1),Integer(2)])
>>> S.max_stable()
{1: 1, 2: 1}
>>> c.dualize()
{1: 0, 2: -1}
>>> S.max_stable() - c == c.dualize()
True

equivalent_recurrent(with_firing_vector=False)[source]¶

Return the recurrent configuration equivalent to the given configuration.

Optionally, this returns the corresponding firing vector.

INPUT:

with_firing_vector – boolean (default: False)

OUTPUT: SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: c = SandpileConfig(S, [0,0,0])
sage: c.equivalent_recurrent() == S.identity()
True
sage: x = c.equivalent_recurrent(True)
sage: r = vector([x[0][v] for v in S.nonsink_vertices()])
sage: f = vector([x[1][v] for v in S.nonsink_vertices()])
sage: cv = vector(c.values())
sage: r == cv - f*S.reduced_laplacian()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> c = SandpileConfig(S, [Integer(0),Integer(0),Integer(0)])
>>> c.equivalent_recurrent() == S.identity()
True
>>> x = c.equivalent_recurrent(True)
>>> r = vector([x[Integer(0)][v] for v in S.nonsink_vertices()])
>>> f = vector([x[Integer(1)][v] for v in S.nonsink_vertices()])
>>> cv = vector(c.values())
>>> r == cv - f*S.reduced_laplacian()
True

Note

Let \(L\) be the reduced Laplacian, \(c\) the initial configuration, \(r\) the returned configuration, and \(f\) the firing vector. Then \(r = c - f\cdot L\).

equivalent_superstable(with_firing_vector=False)[source]¶

Return the equivalent superstable configuration.

Optionally, this returns the corresponding firing vector.

INPUT:

with_firing_vector – boolean (default: False)

OUTPUT: SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: m = S.max_stable()
sage: m.equivalent_superstable().is_superstable()
True
sage: x = m.equivalent_superstable(True)
sage: s = vector(x[0].values())
sage: f = vector(x[1].values())
sage: mv = vector(m.values())
sage: s == mv - f*S.reduced_laplacian()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> m = S.max_stable()
>>> m.equivalent_superstable().is_superstable()
True
>>> x = m.equivalent_superstable(True)
>>> s = vector(x[Integer(0)].values())
>>> f = vector(x[Integer(1)].values())
>>> mv = vector(m.values())
>>> s == mv - f*S.reduced_laplacian()
True

Note

Let \(L\) be the reduced Laplacian, \(c\) the initial configuration, \(s\) the returned configuration, and \(f\) the firing vector. Then \(s = c - f\cdot L\).

fire_script(sigma)[source]¶

Fire the given script.

In other words, fire each vertex the number of times indicated by sigma.

INPUT:

sigma – SandpileConfig or (list or dict representing a SandpileConfig)

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.unstable()
[2, 3]
sage: c.fire_script(SandpileConfig(S,[0,1,1]))
{1: 2, 2: 1, 3: 2}
sage: c.fire_script(SandpileConfig(S,[2,0,0])) == c.fire_vertex(1).fire_vertex(1)
True

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(4))
>>> c = SandpileConfig(S, [Integer(1),Integer(2),Integer(3)])
>>> c.unstable()
[2, 3]
>>> c.fire_script(SandpileConfig(S,[Integer(0),Integer(1),Integer(1)]))
{1: 2, 2: 1, 3: 2}
>>> c.fire_script(SandpileConfig(S,[Integer(2),Integer(0),Integer(0)])) == c.fire_vertex(Integer(1)).fire_vertex(Integer(1))
True

fire_unstable()[source]¶

Fire all unstable vertices.

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.fire_unstable()
{1: 2, 2: 1, 3: 2}

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(4))
>>> c = SandpileConfig(S, [Integer(1),Integer(2),Integer(3)])
>>> c.fire_unstable()
{1: 2, 2: 1, 3: 2}

fire_vertex(v)[source]¶

Fire the given vertex.

INPUT:

v – vertex

OUTPUT: SandpileConfig

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: c.fire_vertex(2)
{1: 2, 2: 0}

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> c = SandpileConfig(S, [Integer(1),Integer(2)])
>>> c.fire_vertex(Integer(2))
{1: 2, 2: 0}

static help(verbose=True)[source]¶

List of SandpileConfig methods.

If verbose, include short descriptions.

INPUT:

verbose – boolean (default: True)

OUTPUT: printed string

EXAMPLES:

Sage

sage: SandpileConfig.help()
Shortcuts for SandpileConfig operations:
~c    -- stabilize
c & d -- add and stabilize
c * c -- add and find equivalent recurrent
c^k   -- add k times and find equivalent recurrent
         (taking inverse if k is negative)

For detailed help with any method FOO listed below,
enter "SandpileConfig.FOO?" or enter "c.FOO?" for any SandpileConfig c.

add_random             -- Add one grain of sand to a random vertex.
burst_size             -- Return the burst size of the configuration with respect to the given
vertex.
deg                    -- The degree of the configuration.
dualize                -- The difference with the maximal stable configuration.
equivalent_recurrent   -- Return the recurrent configuration equivalent to the given
configuration.
equivalent_superstable -- Return the equivalent superstable configuration.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileConfig methods.
is_recurrent           -- Return whether the configuration is recurrent.
is_stable              -- Return whether the configuration is stable.
is_superstable         -- Return whether the configuration is superstable.
is_symmetric           -- Return whether the configuration is symmetric.
order                  -- The order of the equivalent recurrent element.
sandpile               -- The configuration's underlying sandpile.
show                   -- Show the configuration.
stabilize              -- Return the stabilized configuration.
support                -- Return the vertices containing sand.
unstable               -- The unstable vertices.
values                 -- The values of the configuration as a list.

Python

>>> from sage.all import *
>>> SandpileConfig.help()
Shortcuts for SandpileConfig operations:
~c    -- stabilize
c & d -- add and stabilize
c * c -- add and find equivalent recurrent
c^k   -- add k times and find equivalent recurrent
         (taking inverse if k is negative)
<BLANKLINE>
For detailed help with any method FOO listed below,
enter "SandpileConfig.FOO?" or enter "c.FOO?" for any SandpileConfig c.
<BLANKLINE>
add_random             -- Add one grain of sand to a random vertex.
burst_size             -- Return the burst size of the configuration with respect to the given
vertex.
deg                    -- The degree of the configuration.
dualize                -- The difference with the maximal stable configuration.
equivalent_recurrent   -- Return the recurrent configuration equivalent to the given
configuration.
equivalent_superstable -- Return the equivalent superstable configuration.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileConfig methods.
is_recurrent           -- Return whether the configuration is recurrent.
is_stable              -- Return whether the configuration is stable.
is_superstable         -- Return whether the configuration is superstable.
is_symmetric           -- Return whether the configuration is symmetric.
order                  -- The order of the equivalent recurrent element.
sandpile               -- The configuration's underlying sandpile.
show                   -- Show the configuration.
stabilize              -- Return the stabilized configuration.
support                -- Return the vertices containing sand.
unstable               -- The unstable vertices.
values                 -- The values of the configuration as a list.

is_recurrent()[source]¶

Return whether the configuration is recurrent.

OUTPUT: boolean

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.identity().is_recurrent()
True
sage: S.zero_config().is_recurrent()
False

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.identity().is_recurrent()
True
>>> S.zero_config().is_recurrent()
False

is_stable()[source]¶

Return whether the configuration is stable.

OUTPUT: boolean

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.max_stable().is_stable()
True
sage: (2*S.max_stable()).is_stable()
False
sage: (S.max_stable() & S.max_stable()).is_stable()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.max_stable().is_stable()
True
>>> (Integer(2)*S.max_stable()).is_stable()
False
>>> (S.max_stable() & S.max_stable()).is_stable()
True

is_superstable()[source]¶

Return whether the configuration is superstable.

OUTPUT: boolean

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: S.zero_config().is_superstable()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> S.zero_config().is_superstable()
True

is_symmetric(orbits)[source]¶

Return whether the configuration is symmetric.

This returns True if the values of the configuration are constant over the vertices in each sublist of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT: boolean

EXAMPLES:

Sage

sage: S = Sandpile({0: {},
....:              1: {0: 1, 2: 1, 3: 1},
....:              2: {1: 1, 3: 1, 4: 1},
....:              3: {1: 1, 2: 1, 4: 1},
....:              4: {2: 1, 3: 1}})
sage: c = SandpileConfig(S, [1, 2, 2, 3])
sage: c.is_symmetric([[2,3]])
True

Python

>>> from sage.all import *
>>> S = Sandpile({Integer(0): {},
...              Integer(1): {Integer(0): Integer(1), Integer(2): Integer(1), Integer(3): Integer(1)},
...              Integer(2): {Integer(1): Integer(1), Integer(3): Integer(1), Integer(4): Integer(1)},
...              Integer(3): {Integer(1): Integer(1), Integer(2): Integer(1), Integer(4): Integer(1)},
...              Integer(4): {Integer(2): Integer(1), Integer(3): Integer(1)}})
>>> c = SandpileConfig(S, [Integer(1), Integer(2), Integer(2), Integer(3)])
>>> c.is_symmetric([[Integer(2),Integer(3)]])
True

order()[source]¶

The order of the equivalent recurrent element.

OUTPUT: integer

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: c = SandpileConfig(S,[2,0,1])
sage: c.order()
4
sage: ~(c + c + c + c) == S.identity()
True
sage: c = SandpileConfig(S,[1,1,0])
sage: c.order()
1
sage: c.is_recurrent()
False
sage: c.equivalent_recurrent() == S.identity()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> c = SandpileConfig(S,[Integer(2),Integer(0),Integer(1)])
>>> c.order()
4
>>> ~(c + c + c + c) == S.identity()
True
>>> c = SandpileConfig(S,[Integer(1),Integer(1),Integer(0)])
>>> c.order()
1
>>> c.is_recurrent()
False
>>> c.equivalent_recurrent() == S.identity()
True

sandpile()[source]¶

The configuration’s underlying sandpile.

OUTPUT: Sandpile

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
sage: c.sandpile() == S
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> c = S.identity()
>>> c.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
>>> c.sandpile() == S
True

show(sink=True, colors=True, heights=False, directed=None, **kwds)[source]¶

Show the configuration.

INPUT:

sink – boolean (default: True); whether to show the sink
colors – boolean (default: True); whether to color-code the amount of sand on each vertex
heights – boolean (default: False); whether to label each vertex with the amount of sand
directed – (optional) whether to draw directed edges
kwds – (optional) arguments passed to the show method for Graph

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c.show()
sage: c.show(directed=False)
sage: c.show(sink=False, colors=False, heights=True)

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> c = S.identity()
>>> c.show()
>>> c.show(directed=False)
>>> c.show(sink=False, colors=False, heights=True)

stabilize(with_firing_vector=False)[source]¶

Return the stabilized configuration.

Optionally this returns the corresponding firing vector.

INPUT:

with_firing_vector – boolean (default: False)

OUTPUT: SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: c = 2*S.max_stable()
sage: c._set_stabilize()
sage: '_stabilize' in c.__dict__
True
sage: S = sandpiles.House()
sage: c = S.max_stable() + S.identity()
sage: c.stabilize(True)
[{1: 1, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 3, 4: 3}]
sage: S.max_stable() & S.identity() == c.stabilize()
True
sage: ~c == c.stabilize()
True

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> c = Integer(2)*S.max_stable()
>>> c._set_stabilize()
>>> '_stabilize' in c.__dict__
True
>>> S = sandpiles.House()
>>> c = S.max_stable() + S.identity()
>>> c.stabilize(True)
[{1: 1, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 3, 4: 3}]
>>> S.max_stable() & S.identity() == c.stabilize()
True
>>> ~c == c.stabilize()
True

support()[source]¶

Return the vertices containing sand.

OUTPUT: list - support of the configuration

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c
{1: 2, 2: 2, 3: 0}
sage: c.support()
[1, 2]

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> c = S.identity()
>>> c
{1: 2, 2: 2, 3: 0}
>>> c.support()
[1, 2]

unstable()[source]¶

The unstable vertices.

OUTPUT: list of vertices

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.unstable()
[2, 3]

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(4))
>>> c = SandpileConfig(S, [Integer(1),Integer(2),Integer(3)])
>>> c.unstable()
[2, 3]

values()[source]¶

The values of the configuration as a list.

The list is sorted in the order of the vertices.

OUTPUT: list of integers

EXAMPLES:

Sage

sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
sage: c = SandpileConfig(S, {'b':1, 'c':2})
sage: c
{'b': 1, 'c': 2}
sage: c.values()
[1, 2]
sage: S.nonsink_vertices()
['b', 'c']

Python

>>> from sage.all import *
>>> S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
>>> c = SandpileConfig(S, {'b':Integer(1), 'c':Integer(2)})
>>> c
{'b': 1, 'c': 2}
>>> c.values()
[1, 2]
>>> S.nonsink_vertices()
['b', 'c']

class sage.sandpiles.sandpile.SandpileDivisor(S, D)[source]¶

Bases: dict

Class for divisors on a sandpile.

Dcomplex()[source]¶

The support-complex. (See NOTE.)

OUTPUT: simplicial complex

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: p = SandpileDivisor(S, [1,2,1,0,0]).Dcomplex()
sage: p.homology()
{0: 0, 1: Z x Z, 2: 0}
sage: p.f_vector()
[1, 5, 10, 4]
sage: p.betti()
{0: 1, 1: 2, 2: 0}

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> p = SandpileDivisor(S, [Integer(1),Integer(2),Integer(1),Integer(0),Integer(0)]).Dcomplex()
>>> p.homology()
{0: 0, 1: Z x Z, 2: 0}
>>> p.f_vector()
[1, 5, 10, 4]
>>> p.betti()
{0: 1, 1: 2, 2: 0}

Note

The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.

add_random(distrib=None)[source]¶

Add one grain of sand to a random vertex.

INPUT:

distrib – (optional) list of nonnegative numbers representing a probability distribution on the vertices

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = s.zero_div()
sage: D.add_random() # random
{0: 0, 1: 0, 2: 1, 3: 0}
sage: D.add_random([0.1,0.1,0.1,0.7]) # random
{0: 0, 1: 0, 2: 0, 3: 1}

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = s.zero_div()
>>> D.add_random() # random
{0: 0, 1: 0, 2: 1, 3: 0}
>>> D.add_random([RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.1'),RealNumber('0.7')]) # random
{0: 0, 1: 0, 2: 0, 3: 1}

Warning

If distrib is not None, the user is responsible for assuring the sum of its entries is 1.

betti()[source]¶

The Betti numbers for the support-complex. (See NOTE.)

OUTPUT: dictionary of integers

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [2,0,1])
sage: D.betti()
{0: 1, 1: 1}

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(2),Integer(0),Integer(1)])
>>> D.betti()
{0: 1, 1: 1}

Note

The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.

deg()[source]¶

The degree of the divisor.

OUTPUT: integer

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.deg()
6

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.deg()
6

dualize()[source]¶

The difference with the maximal stable divisor.

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.dualize()
{0: 0, 1: -1, 2: -2}
sage: S.max_stable_div() - D == D.dualize()
True

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.dualize()
{0: 0, 1: -1, 2: -2}
>>> S.max_stable_div() - D == D.dualize()
True

effective_div(verbose=True, with_firing_vectors=False)[source]¶

Return all linearly equivalent effective divisors.

If verbose is False, the divisors are converted to lists of integers. If with_firing_vectors is True then a list of firing vectors is also given, each of which prescribes the vertices to be fired in order to obtain an effective divisor.

INPUT:

verbose – boolean (default: True)
with_firing_vectors – boolean (default: False)

OUTPUT: list (of divisors)

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: sorted(D.effective_div(), key=str)
[{0: 0, 1: 2, 2: 0, 3: 4},
 {0: 0, 1: 2, 2: 4, 3: 0},
 {0: 0, 1: 6, 2: 0, 3: 0},
 {0: 1, 1: 3, 2: 1, 3: 1},
 {0: 2, 1: 0, 2: 2, 3: 2},
 {0: 4, 1: 2, 2: 0, 3: 0}]
sage: sorted(D.effective_div(False))
[[0, 2, 0, 4],
 [0, 2, 4, 0],
 [0, 6, 0, 0],
 [1, 3, 1, 1],
 [2, 0, 2, 2],
 [4, 2, 0, 0]]
sage: sorted(D.effective_div(with_firing_vectors=True), key=str)
[({0: 0, 1: 2, 2: 0, 3: 4}, (0, -1, -1, -2)),
 ({0: 0, 1: 2, 2: 4, 3: 0}, (0, -1, -2, -1)),
 ({0: 0, 1: 6, 2: 0, 3: 0}, (0, -2, -1, -1)),
 ({0: 1, 1: 3, 2: 1, 3: 1}, (0, -1, -1, -1)),
 ({0: 2, 1: 0, 2: 2, 3: 2}, (0, 0, -1, -1)),
 ({0: 4, 1: 2, 2: 0, 3: 0}, (0, 0, 0, 0))]
sage: a = _[2]
sage: a[0].values()
[0, 6, 0, 0]
sage: vector(D.values()) - s.laplacian()*a[1]
(0, 6, 0, 0)
sage: sorted(D.effective_div(False, True))
[([0, 2, 0, 4], (0, -1, -1, -2)),
 ([0, 2, 4, 0], (0, -1, -2, -1)),
 ([0, 6, 0, 0], (0, -2, -1, -1)),
 ([1, 3, 1, 1], (0, -1, -1, -1)),
 ([2, 0, 2, 2], (0, 0, -1, -1)),
 ([4, 2, 0, 0], (0, 0, 0, 0))]
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.effective_div(False,True)
[]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(4),Integer(2),Integer(0),Integer(0)])
>>> sorted(D.effective_div(), key=str)
[{0: 0, 1: 2, 2: 0, 3: 4},
 {0: 0, 1: 2, 2: 4, 3: 0},
 {0: 0, 1: 6, 2: 0, 3: 0},
 {0: 1, 1: 3, 2: 1, 3: 1},
 {0: 2, 1: 0, 2: 2, 3: 2},
 {0: 4, 1: 2, 2: 0, 3: 0}]
>>> sorted(D.effective_div(False))
[[0, 2, 0, 4],
 [0, 2, 4, 0],
 [0, 6, 0, 0],
 [1, 3, 1, 1],
 [2, 0, 2, 2],
 [4, 2, 0, 0]]
>>> sorted(D.effective_div(with_firing_vectors=True), key=str)
[({0: 0, 1: 2, 2: 0, 3: 4}, (0, -1, -1, -2)),
 ({0: 0, 1: 2, 2: 4, 3: 0}, (0, -1, -2, -1)),
 ({0: 0, 1: 6, 2: 0, 3: 0}, (0, -2, -1, -1)),
 ({0: 1, 1: 3, 2: 1, 3: 1}, (0, -1, -1, -1)),
 ({0: 2, 1: 0, 2: 2, 3: 2}, (0, 0, -1, -1)),
 ({0: 4, 1: 2, 2: 0, 3: 0}, (0, 0, 0, 0))]
>>> a = _[Integer(2)]
>>> a[Integer(0)].values()
[0, 6, 0, 0]
>>> vector(D.values()) - s.laplacian()*a[Integer(1)]
(0, 6, 0, 0)
>>> sorted(D.effective_div(False, True))
[([0, 2, 0, 4], (0, -1, -1, -2)),
 ([0, 2, 4, 0], (0, -1, -2, -1)),
 ([0, 6, 0, 0], (0, -2, -1, -1)),
 ([1, 3, 1, 1], (0, -1, -1, -1)),
 ([2, 0, 2, 2], (0, 0, -1, -1)),
 ([4, 2, 0, 0], (0, 0, 0, 0))]
>>> D = SandpileDivisor(s,[-Integer(1),Integer(0),Integer(0),Integer(0)])
>>> D.effective_div(False,True)
[]

fire_script(sigma)[source]¶

Fire the given script.

In other words, fire each vertex the number of times indicated by sigma.

INPUT:

sigma – SandpileDivisor or (list or dict representing a SandpileDivisor)

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.unstable()
[1, 2]
sage: D.fire_script([0,1,1])
{0: 3, 1: 1, 2: 2}
sage: D.fire_script(SandpileDivisor(S,[2,0,0])) == D.fire_vertex(0).fire_vertex(0)
True

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.unstable()
[1, 2]
>>> D.fire_script([Integer(0),Integer(1),Integer(1)])
{0: 3, 1: 1, 2: 2}
>>> D.fire_script(SandpileDivisor(S,[Integer(2),Integer(0),Integer(0)])) == D.fire_vertex(Integer(0)).fire_vertex(Integer(0))
True

fire_unstable()[source]¶

Fire all unstable vertices.

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.fire_unstable()
{0: 3, 1: 1, 2: 2}

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.fire_unstable()
{0: 3, 1: 1, 2: 2}

fire_vertex(v)[source]¶

Fire the given vertex.

INPUT:

v – vertex

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.fire_vertex(1)
{0: 2, 1: 0, 2: 4}

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.fire_vertex(Integer(1))
{0: 2, 1: 0, 2: 4}

static help(verbose=True)[source]¶

List of SandpileDivisor methods. If verbose, include short descriptions.

INPUT:

verbose – boolean (default: True)

OUTPUT: printed string

EXAMPLES:

Sage

sage: SandpileDivisor.help()
For detailed help with any method FOO listed below,
enter "SandpileDivisor.FOO?" or enter "D.FOO?" for any SandpileDivisor D.

Dcomplex               -- The support-complex.
add_random             -- Add one grain of sand to a random vertex.
betti                  -- The Betti numbers for the support-complex.
deg                    -- The degree of the divisor.
dualize                -- The difference with the maximal stable divisor.
effective_div          -- Return all linearly equivalent effective divisors.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileDivisor methods.
is_alive               -- Return whether the divisor is stabilizable.
is_linearly_equivalent -- Return whether the given divisor is linearly equivalent.
is_q_reduced           -- Return whether the divisor is q-reduced.
is_symmetric           -- Return whether the divisor is symmetric.
is_weierstrass_pt      -- Return whether the given vertex is a Weierstrass point.
polytope               -- Return the polytope determining the complete linear system.
polytope_integer_pts   -- Return the integer points inside divisor's polytope.
q_reduced              -- Return the linearly equivalent q-reduced divisor.
rank                   -- Return the rank of the divisor.
sandpile               -- The divisor's underlying sandpile.
show                   -- Show the divisor.
simulate_threshold     -- Return the first unstabilizable divisor in the closed Markov chain.
stabilize              -- The stabilization of the divisor.
support                -- List of vertices at which the divisor is nonzero.
unstable               -- Return the unstable vertices.
values                 -- The values of the divisor as a list.
weierstrass_div        -- The Weierstrass divisor.
weierstrass_gap_seq    -- The Weierstrass gap sequence at the given vertex.
weierstrass_pts        -- The Weierstrass points (vertices).
weierstrass_rank_seq   -- The Weierstrass rank sequence at the given vertex.

Python

>>> from sage.all import *
>>> SandpileDivisor.help()
For detailed help with any method FOO listed below,
enter "SandpileDivisor.FOO?" or enter "D.FOO?" for any SandpileDivisor D.
<BLANKLINE>
Dcomplex               -- The support-complex.
add_random             -- Add one grain of sand to a random vertex.
betti                  -- The Betti numbers for the support-complex.
deg                    -- The degree of the divisor.
dualize                -- The difference with the maximal stable divisor.
effective_div          -- Return all linearly equivalent effective divisors.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileDivisor methods.
is_alive               -- Return whether the divisor is stabilizable.
is_linearly_equivalent -- Return whether the given divisor is linearly equivalent.
is_q_reduced           -- Return whether the divisor is q-reduced.
is_symmetric           -- Return whether the divisor is symmetric.
is_weierstrass_pt      -- Return whether the given vertex is a Weierstrass point.
polytope               -- Return the polytope determining the complete linear system.
polytope_integer_pts   -- Return the integer points inside divisor's polytope.
q_reduced              -- Return the linearly equivalent q-reduced divisor.
rank                   -- Return the rank of the divisor.
sandpile               -- The divisor's underlying sandpile.
show                   -- Show the divisor.
simulate_threshold     -- Return the first unstabilizable divisor in the closed Markov chain.
stabilize              -- The stabilization of the divisor.
support                -- List of vertices at which the divisor is nonzero.
unstable               -- Return the unstable vertices.
values                 -- The values of the divisor as a list.
weierstrass_div        -- The Weierstrass divisor.
weierstrass_gap_seq    -- The Weierstrass gap sequence at the given vertex.
weierstrass_pts        -- The Weierstrass points (vertices).
weierstrass_rank_seq   -- The Weierstrass rank sequence at the given vertex.

is_alive(cycle=False)[source]¶

Return whether the divisor is stabilizable.

In other words, will the divisor stabilize under repeated firings of all unstable vertices? Optionally this returns the resulting cycle.

INPUT:

cycle – boolean (default: False)

OUTPUT: boolean or optionally, a list of SandpileDivisors

EXAMPLES:

Sage

sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, {0: 4, 1: 3, 2: 3, 3: 2})
sage: D.is_alive()
True
sage: D.is_alive(True)
[{0: 4, 1: 3, 2: 3, 3: 2}, {0: 3, 1: 2, 2: 2, 3: 5}, {0: 1, 1: 4, 2: 4, 3: 3}]

Python

>>> from sage.all import *
>>> S = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(S, {Integer(0): Integer(4), Integer(1): Integer(3), Integer(2): Integer(3), Integer(3): Integer(2)})
>>> D.is_alive()
True
>>> D.is_alive(True)
[{0: 4, 1: 3, 2: 3, 3: 2}, {0: 3, 1: 2, 2: 2, 3: 5}, {0: 1, 1: 4, 2: 4, 3: 3}]

is_linearly_equivalent(D, with_firing_vector=False)[source]¶

Return whether the given divisor is linearly equivalent.

Optionally, this returns the firing vector. (See NOTE.)

INPUT:

D – SandpileDivisor or list, tuple, etc. representing a divisor
with_firing_vector – boolean (default: False)

OUTPUT: boolean or integer vector

EXAMPLES:

Sage

sage: s = sandpiles.Complete(3)
sage: D = SandpileDivisor(s,[2,0,0])
sage: D.is_linearly_equivalent([0,1,1])
True
sage: D.is_linearly_equivalent([0,1,1],True)
(0, -1, -1)
sage: v = vector(D.is_linearly_equivalent([0,1,1],True))
sage: vector(D.values()) - s.laplacian()*v
(0, 1, 1)
sage: D.is_linearly_equivalent([0,0,0])
False
sage: D.is_linearly_equivalent([0,0,0],True)
()

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(3))
>>> D = SandpileDivisor(s,[Integer(2),Integer(0),Integer(0)])
>>> D.is_linearly_equivalent([Integer(0),Integer(1),Integer(1)])
True
>>> D.is_linearly_equivalent([Integer(0),Integer(1),Integer(1)],True)
(0, -1, -1)
>>> v = vector(D.is_linearly_equivalent([Integer(0),Integer(1),Integer(1)],True))
>>> vector(D.values()) - s.laplacian()*v
(0, 1, 1)
>>> D.is_linearly_equivalent([Integer(0),Integer(0),Integer(0)])
False
>>> D.is_linearly_equivalent([Integer(0),Integer(0),Integer(0)],True)
()

Note

If with_firing_vector is False, returns either True or False.
If with_firing_vector is True then: (i) if self is linearly equivalent to \(D\), returns a vector \(v\) such that self - v*self.laplacian().transpose() = D. Otherwise, (ii) if self is not linearly equivalent to \(D\), the output is the empty vector, ().

is_q_reduced()[source]¶

Return whether the divisor is \(q\)-reduced.

This would mean that \(self = c + kq\) where \(c\) is superstable, \(k\) is an integer, and \(q\) is the sink vertex.

OUTPUT: boolean

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[2,-3,2,0])
sage: D.is_q_reduced()
False
sage: SandpileDivisor(s,[10,0,1,2]).is_q_reduced()
True

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(2),-Integer(3),Integer(2),Integer(0)])
>>> D.is_q_reduced()
False
>>> SandpileDivisor(s,[Integer(10),Integer(0),Integer(1),Integer(2)]).is_q_reduced()
True

For undirected or, more generally, Eulerian graphs, \(q\)-reduced divisors are linearly equivalent if and only if they are equal. The same does not hold for general directed graphs:

Sage

sage: s = Sandpile({0:[1],1:[1,1]})
sage: D = SandpileDivisor(s,[-1,1])
sage: Z = s.zero_div()
sage: D.is_q_reduced()
True
sage: Z.is_q_reduced()
True
sage: D == Z
False
sage: D.is_linearly_equivalent(Z)
True

[Python]

>>> from sage.all import *
>>> s = Sandpile({Integer(0):[Integer(1)],Integer(1):[Integer(1),Integer(1)]})
>>> D = SandpileDivisor(s,[-Integer(1),Integer(1)])
>>> Z = s.zero_div()
>>> D.is_q_reduced()
True
>>> Z.is_q_reduced()
True
>>> D == Z
False
>>> D.is_linearly_equivalent(Z)
True

is_symmetric(orbits)[source]¶

Return whether the divisor is symmetric.

This returns True if the values of the configuration are constant over the vertices in each sublist of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT: boolean

EXAMPLES:

Sage

sage: S = sandpiles.House()
sage: S.dict()
{0: {1: 1, 2: 1},
 1: {0: 1, 3: 1},
 2: {0: 1, 3: 1, 4: 1},
 3: {1: 1, 2: 1, 4: 1},
 4: {2: 1, 3: 1}}
sage: D = SandpileDivisor(S, [0,0,1,1,3])
sage: D.is_symmetric([[2,3], [4]])
True

Python

>>> from sage.all import *
>>> S = sandpiles.House()
>>> S.dict()
{0: {1: 1, 2: 1},
 1: {0: 1, 3: 1},
 2: {0: 1, 3: 1, 4: 1},
 3: {1: 1, 2: 1, 4: 1},
 4: {2: 1, 3: 1}}
>>> D = SandpileDivisor(S, [Integer(0),Integer(0),Integer(1),Integer(1),Integer(3)])
>>> D.is_symmetric([[Integer(2),Integer(3)], [Integer(4)]])
True

is_weierstrass_pt(v='sink')[source]¶

Return whether the given vertex is a Weierstrass point.

INPUT:

v – (default: sink) vertex

OUTPUT: boolean

EXAMPLES:

Sage

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: K.weierstrass_rank_seq()  # sequence at the sink vertex, 0
(1, 0, -1)
sage: K.is_weierstrass_pt()
False
sage: K.weierstrass_rank_seq(4)
(1, 0, 0, -1)
sage: K.is_weierstrass_pt(4)
True

Python

>>> from sage.all import *
>>> s = sandpiles.House()
>>> K = s.canonical_divisor()
>>> K.weierstrass_rank_seq()  # sequence at the sink vertex, 0
(1, 0, -1)
>>> K.is_weierstrass_pt()
False
>>> K.weierstrass_rank_seq(Integer(4))
(1, 0, 0, -1)
>>> K.is_weierstrass_pt(Integer(4))
True

Note

The vertex \(v\) is a (generalized) Weierstrass point for divisor \(D\) if the sequence of ranks \(r(D - nv)\) for \(n = 0, 1, 2, \dots\) is not \(r(D), r(D)-1, \dots, 0, -1, -1, \dots\)

polytope()[source]¶

Return the polytope determining the complete linear system.

OUTPUT: polytope

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: p = D.polytope()
sage: p.inequalities()
(An inequality (-3, 1, 1) x + 2 >= 0,
 An inequality (1, 1, 1) x + 4 >= 0,
 An inequality (1, -3, 1) x + 0 >= 0,
 An inequality (1, 1, -3) x + 0 >= 0)
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.polytope()
The empty polyhedron in QQ^3

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(4),Integer(2),Integer(0),Integer(0)])
>>> p = D.polytope()
>>> p.inequalities()
(An inequality (-3, 1, 1) x + 2 >= 0,
 An inequality (1, 1, 1) x + 4 >= 0,
 An inequality (1, -3, 1) x + 0 >= 0,
 An inequality (1, 1, -3) x + 0 >= 0)
>>> D = SandpileDivisor(s,[-Integer(1),Integer(0),Integer(0),Integer(0)])
>>> D.polytope()
The empty polyhedron in QQ^3

Note

For a divisor \(D\), this is the intersection of (i) the polyhedron determined by the system of inequalities \(L^t x \leq D\) where \(L^t\) is the transpose of the Laplacian with (ii) the hyperplane \(x_{\mathrm{sink\_vertex}} = 0\). The polytope is thought of as sitting in \((n-1)\)-dimensional Euclidean space where \(n\) is the number of vertices.

polytope_integer_pts()[source]¶

Return the integer points inside divisor’s polytope.

The polytope referred to here is the one determining the divisor’s complete linear system (see the documentation for polytope).

OUTPUT: tuple of integer vectors

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: sorted(D.polytope_integer_pts())
[(-2, -1, -1),
 (-1, -2, -1),
 (-1, -1, -2),
 (-1, -1, -1),
 (0, -1, -1),
 (0, 0, 0)]
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.polytope_integer_pts()
()

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(4),Integer(2),Integer(0),Integer(0)])
>>> sorted(D.polytope_integer_pts())
[(-2, -1, -1),
 (-1, -2, -1),
 (-1, -1, -2),
 (-1, -1, -1),
 (0, -1, -1),
 (0, 0, 0)]
>>> D = SandpileDivisor(s,[-Integer(1),Integer(0),Integer(0),Integer(0)])
>>> D.polytope_integer_pts()
()

q_reduced(verbose=True)[source]¶

Return the linearly equivalent \(q\)-reduced divisor.

INPUT:

verbose – boolean (default: True)

OUTPUT: SandpileDivisor or list representing SandpileDivisor

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[2,-3,2,0])
sage: D.q_reduced()
{0: -2, 1: 1, 2: 2, 3: 0}
sage: D.q_reduced(False)
[-2, 1, 2, 0]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(2),-Integer(3),Integer(2),Integer(0)])
>>> D.q_reduced()
{0: -2, 1: 1, 2: 2, 3: 0}
>>> D.q_reduced(False)
[-2, 1, 2, 0]

Note

The divisor \(D\) is \(qreduced if `D = c + kq\) where \(c\) is superstable, \(k\) is an integer, and \(q\) is the sink.

rank(with_witness=False)[source]¶

Return the rank of the divisor.

Optionally this returns an effective divisor \(E\) such that \(D - E\) is not winnable (has an empty complete linear system).

INPUT:

with_witness – boolean (default: False)

OUTPUT: integer or (integer, SandpileDivisor)

EXAMPLES:

   sage: S = sandpiles.Complete(4)
   sage: D = SandpileDivisor(S,[4,2,0,0])
   sage: D.rank()
   3
   sage: D.rank(True)
   (3, {0: 3, 1: 0, 2: 1, 3: 0})
   sage: E = _[1]
   sage: (D - E).rank()
   -1

Riemann-Roch theorem::

   sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus()
   True

Riemann-Roch theorem::

   sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus()
   True
   sage: S = Sandpile({0:[1,1,1,2],1:[0,0,0,1,1,1,2,2],2:[2,2,1,1,0]},0)  # multigraph with loops
   sage: D = SandpileDivisor(S,[4,2,0])
   sage: D.rank(True)
   (2, {0: 1, 1: 1, 2: 1})
   sage: S = Sandpile({0:[1,2], 1:[0,2,2], 2: [0,1]},0) # directed graph
   sage: S.is_undirected()
   False
   sage: D = SandpileDivisor(S,[0,2,0])
   sage: D.effective_div()
   [{0: 0, 1: 2, 2: 0}, {0: 2, 1: 0, 2: 0}]
   sage: D.rank(True)
   (0, {0: 0, 1: 0, 2: 1})
   sage: E = D.rank(True)[1]
   sage: (D - E).effective_div()
   []

Note

The rank of a divisor \(D\) is -1 if \(D\) is not linearly equivalent to an effective divisor (i.e., the dollar game represented by \(D\) is unwinnable). Otherwise, the rank of \(D\) is the largest integer \(r\) such that \(D - E\) is linearly equivalent to an effective divisor for all effective divisors \(E\) with \(\deg(E) = r\).

sandpile()[source]¶

The divisor’s underlying sandpile.

OUTPUT: Sandpile

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: D = SandpileDivisor(S,[1,-2,0,3])
sage: D.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
sage: D.sandpile() == S
True

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> D = SandpileDivisor(S,[Integer(1),-Integer(2),Integer(0),Integer(3)])
>>> D.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
>>> D.sandpile() == S
True

show(heights=True, directed=None, **kwds)[source]¶

Show the divisor.

INPUT:

heights – boolean (default: True); whether to label each vertex with the amount of sand
directed – (optional) whether to draw directed edges
kwds – (optional) arguments passed to the show method for Graph

EXAMPLES:

Sage

sage: S = sandpiles.Diamond()
sage: D = SandpileDivisor(S, [1,-2,0,2])
sage: D.show(graph_border=True, vertex_size=700, directed=False)

Python

>>> from sage.all import *
>>> S = sandpiles.Diamond()
>>> D = SandpileDivisor(S, [Integer(1),-Integer(2),Integer(0),Integer(2)])
>>> D.show(graph_border=True, vertex_size=Integer(700), directed=False)

simulate_threshold(distrib=None)[source]¶

Return the first unstabilizable divisor in the closed Markov chain.

(See NOTE.)

INPUT:

distrib – (optional) list of nonnegative numbers representing a probability distribution on the vertices

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = s.zero_div()
sage: D.simulate_threshold()  # random
{0: 2, 1: 3, 2: 1, 3: 2}
sage: n(mean([D.simulate_threshold().deg() for _ in range(10)]))  # random
7.10000000000000
sage: n(s.stationary_density()*s.n_vertices())
6.93750000000000

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = s.zero_div()
>>> D.simulate_threshold()  # random
{0: 2, 1: 3, 2: 1, 3: 2}
>>> n(mean([D.simulate_threshold().deg() for _ in range(Integer(10))]))  # random
7.10000000000000
>>> n(s.stationary_density()*s.n_vertices())
6.93750000000000

Note

Starting at self, repeatedly choose a vertex and add a grain of sand to it. Return the first unstabilizable divisor that is reached. Also see the markov_chain method for the underlying sandpile.

stabilize(with_firing_vector=False)[source]¶

The stabilization of the divisor. If not stabilizable, return an error.

INPUT:

with_firing_vector – boolean (default: False)

EXAMPLES:

Sage

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[0,3,0,0])
sage: D.stabilize()
{0: 1, 1: 0, 2: 1, 3: 1}
sage: D.stabilize(with_firing_vector=True)
[{0: 1, 1: 0, 2: 1, 3: 1}, {0: 0, 1: 1, 2: 0, 3: 0}]

Python

>>> from sage.all import *
>>> s = sandpiles.Complete(Integer(4))
>>> D = SandpileDivisor(s,[Integer(0),Integer(3),Integer(0),Integer(0)])
>>> D.stabilize()
{0: 1, 1: 0, 2: 1, 3: 1}
>>> D.stabilize(with_firing_vector=True)
[{0: 1, 1: 0, 2: 1, 3: 1}, {0: 0, 1: 1, 2: 0, 3: 0}]

support()[source]¶

List of vertices at which the divisor is nonzero.

OUTPUT: list representing the support of the divisor

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(4)
sage: D = SandpileDivisor(S, [0,0,1,1])
sage: D.support()
[2, 3]
sage: S.vertices(sort=True)
[0, 1, 2, 3]

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(4))
>>> D = SandpileDivisor(S, [Integer(0),Integer(0),Integer(1),Integer(1)])
>>> D.support()
[2, 3]
>>> S.vertices(sort=True)
[0, 1, 2, 3]

unstable()[source]¶

Return the unstable vertices.

OUTPUT: list of vertices

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.unstable()
[1, 2]

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(3))
>>> D = SandpileDivisor(S, [Integer(1),Integer(2),Integer(3)])
>>> D.unstable()
[1, 2]

values()[source]¶

The values of the divisor as a list.

The list is sorted in the order of the vertices.

OUTPUT: list of integers

boolean

EXAMPLES:

Sage

sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
sage: D = SandpileDivisor(S, {'a':0, 'b':1, 'c':2})
sage: D
{'a': 0, 'b': 1, 'c': 2}
sage: D.values()
[0, 1, 2]
sage: S.vertices(sort=True)
['a', 'b', 'c']

Python

>>> from sage.all import *
>>> S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
>>> D = SandpileDivisor(S, {'a':Integer(0), 'b':Integer(1), 'c':Integer(2)})
>>> D
{'a': 0, 'b': 1, 'c': 2}
>>> D.values()
[0, 1, 2]
>>> S.vertices(sort=True)
['a', 'b', 'c']

weierstrass_div(verbose=True)[source]¶

The Weierstrass divisor. Its value at a vertex is the weight of that vertex as a Weierstrass point. (See SandpileDivisor.weierstrass_gap_seq.)

INPUT:

verbose – boolean (default: True)

OUTPUT: SandpileDivisor

EXAMPLES:

Sage

sage: s = sandpiles.Diamond()
sage: D = SandpileDivisor(s,[4,2,1,0])
sage: [D.weierstrass_rank_seq(v) for v in s]
[(5, 4, 3, 2, 1, 0, 0, -1),
 (5, 4, 3, 2, 1, 0, -1),
 (5, 4, 3, 2, 1, 0, 0, 0, -1),
 (5, 4, 3, 2, 1, 0, 0, -1)]
sage: D.weierstrass_div()
{0: 1, 1: 0, 2: 2, 3: 1}
sage: k5 = sandpiles.Complete(5)
sage: K = k5.canonical_divisor()
sage: K.weierstrass_div()
{0: 9, 1: 9, 2: 9, 3: 9, 4: 9}

Python

>>> from sage.all import *
>>> s = sandpiles.Diamond()
>>> D = SandpileDivisor(s,[Integer(4),Integer(2),Integer(1),Integer(0)])
>>> [D.weierstrass_rank_seq(v) for v in s]
[(5, 4, 3, 2, 1, 0, 0, -1),
 (5, 4, 3, 2, 1, 0, -1),
 (5, 4, 3, 2, 1, 0, 0, 0, -1),
 (5, 4, 3, 2, 1, 0, 0, -1)]
>>> D.weierstrass_div()
{0: 1, 1: 0, 2: 2, 3: 1}
>>> k5 = sandpiles.Complete(Integer(5))
>>> K = k5.canonical_divisor()
>>> K.weierstrass_div()
{0: 9, 1: 9, 2: 9, 3: 9, 4: 9}

weierstrass_gap_seq(v='sink', weight=True)[source]¶

The Weierstrass gap sequence at the given vertex. If weight is True, then also compute the weight of each gap value.

INPUT:

v – (default: sink) vertex
weight – boolean (default: True)

OUTPUT: list or (list of list) of integers

EXAMPLES:

Sage

sage: s = sandpiles.Cycle(4)
sage: D = SandpileDivisor(s,[2,0,0,0])
sage: [D.weierstrass_gap_seq(v,False) for v in s.vertices(sort=True)]
[(1, 3), (1, 2), (1, 3), (1, 2)]
sage: [D.weierstrass_gap_seq(v) for v in s.vertices(sort=True)]
[((1, 3), 1), ((1, 2), 0), ((1, 3), 1), ((1, 2), 0)]
sage: D.weierstrass_gap_seq()   # gap sequence at sink vertex, 0
((1, 3), 1)
sage: D.weierstrass_rank_seq()  # rank sequence at the sink vertex
(1, 0, 0, -1)

Python

>>> from sage.all import *
>>> s = sandpiles.Cycle(Integer(4))
>>> D = SandpileDivisor(s,[Integer(2),Integer(0),Integer(0),Integer(0)])
>>> [D.weierstrass_gap_seq(v,False) for v in s.vertices(sort=True)]
[(1, 3), (1, 2), (1, 3), (1, 2)]
>>> [D.weierstrass_gap_seq(v) for v in s.vertices(sort=True)]
[((1, 3), 1), ((1, 2), 0), ((1, 3), 1), ((1, 2), 0)]
>>> D.weierstrass_gap_seq()   # gap sequence at sink vertex, 0
((1, 3), 1)
>>> D.weierstrass_rank_seq()  # rank sequence at the sink vertex
(1, 0, 0, -1)

Note

The integer \(k\) is a Weierstrass gap for the divisor \(D\) at vertex \(v\) if the rank of \(D - (k-1)v\) does not equal the rank of \(D - kv\). Let \(r\) be the rank of \(D\) and let \(k_i\) be the \(i\)-th gap at \(v\). The Weierstrass weight of \(v\) for \(D\) is the sum of \((k_i - i)\) as \(i\) ranges from \(1\) to \(r + 1\). It measure the difference between the sequence \(r, r - 1, ..., 0, -1, -1, ...\) and the rank sequence \(\mathrm{rank}(D), \mathrm{rank}(D - v), \mathrm{rank}(D - 2v), \dots\)

weierstrass_pts(with_rank_seq=False)[source]¶

The Weierstrass points (vertices). Optionally, return the corresponding rank sequences.

INPUT:

with_rank_seq – boolean (default: False)

OUTPUT: tuple of vertices or list of (vertex, rank sequence)

EXAMPLES:

Sage

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: K.weierstrass_pts()
(4,)
sage: K.weierstrass_pts(True)
[(4, (1, 0, 0, -1))]

Python

>>> from sage.all import *
>>> s = sandpiles.House()
>>> K = s.canonical_divisor()
>>> K.weierstrass_pts()
(4,)
>>> K.weierstrass_pts(True)
[(4, (1, 0, 0, -1))]

Note

The vertex \(v\) is a (generalized) Weierstrass point for divisor \(D\) if the sequence of ranks \(r(D - nv)\) for \(n = 0, 1, 2, \dots`\) is not \(r(D), r(D)-1, \dots, 0, -1, -1, \dots\)

weierstrass_rank_seq(v='sink')[source]¶

The Weierstrass rank sequence at the given vertex. Computes the rank of the divisor \(D - nv\) starting with \(n=0\) and ending when the rank is \(-1\).

INPUT:

v – (default: sink) vertex

OUTPUT: tuple of int

EXAMPLES:

Sage

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: [K.weierstrass_rank_seq(v) for v in s.vertices(sort=True)]
[(1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, 0, -1)]

Python

>>> from sage.all import *
>>> s = sandpiles.House()
>>> K = s.canonical_divisor()
>>> [K.weierstrass_rank_seq(v) for v in s.vertices(sort=True)]
[(1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, 0, -1)]

sage.sandpiles.sandpile.admissible_partitions(S, k)[source]¶

The partitions of the vertices of \(S\) into \(k\) parts, each of which is connected.

INPUT:

S – Sandpile
k – integer

OUTPUT: partitions

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import admissible_partitions
sage: from sage.sandpiles.sandpile import partition_sandpile
sage: S = sandpiles.Cycle(4)
sage: P = [list(admissible_partitions(S, i)) for i in [2,3,4]]
sage: P
[[{{0, 2, 3}, {1}},
  {{0, 3}, {1, 2}},
  {{0, 1, 3}, {2}},
  {{0}, {1, 2, 3}},
  {{0, 1}, {2, 3}},
  {{0, 1, 2}, {3}}],
 [{{0, 3}, {1}, {2}},
  {{0}, {1}, {2, 3}},
  {{0}, {1, 2}, {3}},
  {{0, 1}, {2}, {3}}],
 [{{0}, {1}, {2}, {3}}]]
sage: for p in P:
....:  sum([partition_sandpile(S, i).betti(verbose=False)[-1] for i in p])
6
8
3
sage: S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     6     8     3
------------------------------
total:     1     6     8     3

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import admissible_partitions
>>> from sage.sandpiles.sandpile import partition_sandpile
>>> S = sandpiles.Cycle(Integer(4))
>>> P = [list(admissible_partitions(S, i)) for i in [Integer(2),Integer(3),Integer(4)]]
>>> P
[[{{0, 2, 3}, {1}},
  {{0, 3}, {1, 2}},
  {{0, 1, 3}, {2}},
  {{0}, {1, 2, 3}},
  {{0, 1}, {2, 3}},
  {{0, 1, 2}, {3}}],
 [{{0, 3}, {1}, {2}},
  {{0}, {1}, {2, 3}},
  {{0}, {1, 2}, {3}},
  {{0, 1}, {2}, {3}}],
 [{{0}, {1}, {2}, {3}}]]
>>> for p in P:
...  sum([partition_sandpile(S, i).betti(verbose=False)[-Integer(1)] for i in p])
6
8
3
>>> S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     6     8     3
------------------------------
total:     1     6     8     3

sage.sandpiles.sandpile.aztec_sandpile(n)[source]¶

The aztec diamond graph.

INPUT:

n – integer

OUTPUT: dictionary for the aztec diamond graph

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import aztec_sandpile
sage: T = aztec_sandpile(2)
sage: sorted(len(v) for u, v in T.items())
[3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 8]
sage: Sandpile(T,(0, 0)).group_order()
4542720

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import aztec_sandpile
>>> T = aztec_sandpile(Integer(2))
>>> sorted(len(v) for u, v in T.items())
[3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 8]
>>> Sandpile(T,(Integer(0), Integer(0))).group_order()
4542720

Note

This is the aztec diamond graph with a sink vertex added. Boundary vertices have edges to the sink so that each vertex has degree 4.

sage.sandpiles.sandpile.firing_graph(S, eff)[source]¶

Create a digraph with divisors as vertices and edges between two divisors \(D\) and \(E\) if firing a single vertex in \(D\) gives \(E\).

INPUT:

S – Sandpile
eff – list of divisors

OUTPUT: DiGraph

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(6)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div()
sage: firing_graph(S, eff).show3d(edge_size=.005,               # long time
....:                             vertex_size=0.01)

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(6))
>>> D = SandpileDivisor(S, [Integer(1),Integer(1),Integer(1),Integer(1),Integer(2),Integer(0)])
>>> eff = D.effective_div()
>>> firing_graph(S, eff).show3d(edge_size=RealNumber('.005'),               # long time
...                             vertex_size=RealNumber('0.01'))

sage.sandpiles.sandpile.glue_graphs(g, h, glue_g, glue_h)[source]¶

Glue two graphs together.

INPUT:

g, h – dictionaries for directed multigraphs
glue_h, glue_g – dictionaries for a vertex

OUTPUT: dictionary for a directed multigraph

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import glue_graphs
sage: x = {0: {}, 1: {0: 1}, 2: {0: 1, 1: 1}, 3: {0: 1, 1: 1, 2: 1}}
sage: y = {0: {}, 1: {0: 2}, 2: {1: 2}, 3: {0: 1, 2: 1}}
sage: glue_x = {1: 1, 3: 2}
sage: glue_y = {0: 1, 1: 2, 3: 1}
sage: z = glue_graphs(x,y,glue_x,glue_y); z
{'sink': {},
 'x0': {'sink': 1, 'x1': 1, 'x3': 2, 'y1': 2, 'y3': 1},
 'x1': {'x0': 1},
 'x2': {'x0': 1, 'x1': 1},
 'x3': {'x0': 1, 'x1': 1, 'x2': 1},
 'y1': {'sink': 2},
 'y2': {'y1': 2},
 'y3': {'sink': 1, 'y2': 1}}
sage: S = Sandpile(z,'sink')
sage: S.h_vector()
[1, 6, 17, 31, 41, 41, 31, 17, 6, 1]
sage: S.resolution()
'R^1 <-- R^7 <-- R^21 <-- R^35 <-- R^35 <-- R^21 <-- R^7 <-- R^1'

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import glue_graphs
>>> x = {Integer(0): {}, Integer(1): {Integer(0): Integer(1)}, Integer(2): {Integer(0): Integer(1), Integer(1): Integer(1)}, Integer(3): {Integer(0): Integer(1), Integer(1): Integer(1), Integer(2): Integer(1)}}
>>> y = {Integer(0): {}, Integer(1): {Integer(0): Integer(2)}, Integer(2): {Integer(1): Integer(2)}, Integer(3): {Integer(0): Integer(1), Integer(2): Integer(1)}}
>>> glue_x = {Integer(1): Integer(1), Integer(3): Integer(2)}
>>> glue_y = {Integer(0): Integer(1), Integer(1): Integer(2), Integer(3): Integer(1)}
>>> z = glue_graphs(x,y,glue_x,glue_y); z
{'sink': {},
 'x0': {'sink': 1, 'x1': 1, 'x3': 2, 'y1': 2, 'y3': 1},
 'x1': {'x0': 1},
 'x2': {'x0': 1, 'x1': 1},
 'x3': {'x0': 1, 'x1': 1, 'x2': 1},
 'y1': {'sink': 2},
 'y2': {'y1': 2},
 'y3': {'sink': 1, 'y2': 1}}
>>> S = Sandpile(z,'sink')
>>> S.h_vector()
[1, 6, 17, 31, 41, 41, 31, 17, 6, 1]
>>> S.resolution()
'R^1 <-- R^7 <-- R^21 <-- R^35 <-- R^35 <-- R^21 <-- R^7 <-- R^1'

Note

This method makes a dictionary for a graph by combining those for \(g\) and \(h\). The sink of \(g\) is replaced by a vertex that is connected to the vertices of \(g\) as specified by glue_g the vertices of \(h\) as specified in glue_h. The sink of the glued graph is 'sink'.

Both glue_g and glue_h are dictionaries with entries of the form v:w where v is the vertex to be connected to and w is the weight of the connecting edge.

sage.sandpiles.sandpile.min_cycles(G, v)[source]¶

Minimal length cycles in the digraph \(G\) starting at vertex \(v\).

INPUT:

G – DiGraph
v – vertex of G

OUTPUT: list of lists of vertices

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import min_cycles, sandlib
sage: T = sandlib('gor')
sage: [min_cycles(T, i) for i in T.vertices(sort=True)]
[[], [[1, 3]], [[2, 3, 1], [2, 3]], [[3, 1], [3, 2]]]

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import min_cycles, sandlib
>>> T = sandlib('gor')
>>> [min_cycles(T, i) for i in T.vertices(sort=True)]
[[], [[1, 3]], [[2, 3, 1], [2, 3]], [[3, 1], [3, 2]]]

sage.sandpiles.sandpile.parallel_firing_graph(S, eff)[source]¶

Create a digraph with divisors as vertices and edges between two divisors \(D\) and \(E\) if firing all unstable vertices in \(D\) gives \(E\).

INPUT:

S – Sandpile
eff – list of divisors

OUTPUT: DiGraph

EXAMPLES:

Sage

sage: S = sandpiles.Cycle(6)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div()
sage: parallel_firing_graph(S, eff).show3d(edge_size=.005,      # long time
....:                                      vertex_size=0.01)

Python

>>> from sage.all import *
>>> S = sandpiles.Cycle(Integer(6))
>>> D = SandpileDivisor(S, [Integer(1),Integer(1),Integer(1),Integer(1),Integer(2),Integer(0)])
>>> eff = D.effective_div()
>>> parallel_firing_graph(S, eff).show3d(edge_size=RealNumber('.005'),      # long time
...                                      vertex_size=RealNumber('0.01'))

sage.sandpiles.sandpile.partition_sandpile(S, p)[source]¶

Each set of vertices in \(p\) is regarded as a single vertex, with and edge between \(A\) and \(B\) if some element of \(A\) is connected by an edge to some element of \(B\) in \(S\).

INPUT:

S – Sandpile
p – partition of the vertices of S

OUTPUT: Sandpile

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import admissible_partitions, partition_sandpile
sage: S = sandpiles.Cycle(4)
sage: P = [list(admissible_partitions(S, i)) for i in [2,3,4]]
sage: for p in P:
....:  sum([partition_sandpile(S, i).betti(verbose=False)[-1] for i in p])
6
8
3
sage: S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     6     8     3
------------------------------
total:     1     6     8     3

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import admissible_partitions, partition_sandpile
>>> S = sandpiles.Cycle(Integer(4))
>>> P = [list(admissible_partitions(S, i)) for i in [Integer(2),Integer(3),Integer(4)]]
>>> for p in P:
...  sum([partition_sandpile(S, i).betti(verbose=False)[-Integer(1)] for i in p])
6
8
3
>>> S.betti()
           0     1     2     3
------------------------------
    0:     1     -     -     -
    1:     -     6     8     3
------------------------------
total:     1     6     8     3

sage.sandpiles.sandpile.sandlib(selector=None)[source]¶

Return the sandpile identified by selector. If no argument is given, a description of the sandpiles in the sandlib is printed.

INPUT:

selector – (optional) identifier or None

OUTPUT: Sandpile or description

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import sandlib
sage: sandlib()
  Sandpiles in the sandlib:
     ci1 : complete intersection, non-DAG but equivalent to a DAG
     generic : generic digraph with 6 vertices
     genus2 : Undirected graph of genus 2
     gor : Gorenstein but not a complete intersection
     kite : generic undirected graphs with 5 vertices
     riemann-roch1 : directed graph with postulation 9 and 3 maximal weight superstables
     riemann-roch2 : directed graph with a superstable not majorized by a maximal superstable
sage: S = sandlib('gor')
sage: S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import sandlib
>>> sandlib()
  Sandpiles in the sandlib:
     ci1 : complete intersection, non-DAG but equivalent to a DAG
     generic : generic digraph with 6 vertices
     genus2 : Undirected graph of genus 2
     gor : Gorenstein but not a complete intersection
     kite : generic undirected graphs with 5 vertices
     riemann-roch1 : directed graph with postulation 9 and 3 maximal weight superstables
     riemann-roch2 : directed graph with a superstable not majorized by a maximal superstable
>>> S = sandlib('gor')
>>> S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'

sage.sandpiles.sandpile.triangle_sandpile(n)[source]¶

A triangular sandpile. Each nonsink vertex has out-degree six. The vertices on the boundary of the triangle are connected to the sink.

INPUT:

n – integer

OUTPUT: Sandpile

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import triangle_sandpile
sage: T = triangle_sandpile(5)
sage: T.group_order()
135418115000

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import triangle_sandpile
>>> T = triangle_sandpile(Integer(5))
>>> T.group_order()
135418115000

sage.sandpiles.sandpile.wilmes_algorithm(M)[source]¶

Compute an integer matrix \(L\) with the same integer row span as \(M\) and such that \(L\) is the reduced Laplacian of a directed multigraph.

INPUT:

M – square integer matrix of full rank

OUTPUT: integer matrix (L)

EXAMPLES:

Sage

sage: from sage.sandpiles.sandpile import wilmes_algorithm
sage: P = matrix([[2,3,-7,-3],[5,2,-5,5],[8,2,5,4],[-5,-9,6,6]])
sage: wilmes_algorithm(P)
[ 3279   -79 -1599 -1600]
[   -1  1539  -136 -1402]
[    0    -1  1650 -1649]
[    0     0 -1658  1658]

Python

>>> from sage.all import *
>>> from sage.sandpiles.sandpile import wilmes_algorithm
>>> P = matrix([[Integer(2),Integer(3),-Integer(7),-Integer(3)],[Integer(5),Integer(2),-Integer(5),Integer(5)],[Integer(8),Integer(2),Integer(5),Integer(4)],[-Integer(5),-Integer(9),Integer(6),Integer(6)]])
>>> wilmes_algorithm(P)
[ 3279   -79 -1599 -1600]
[   -1  1539  -136 -1402]
[    0    -1  1650 -1649]
[    0     0 -1658  1658]

REFERENCES:

[PPW2013]