Sage Sandpiles is a package for calculations involving Dhar’s abelian sandpile
model (ASM) using the open-source mathematics software, Sage. A brief
introduction to the ASM follows. For a more thorough introduction, the paper *Chip-Firing and Rotor-Routing on Directed Graphs* [H], by Holroyd et al. is recommended.

To describe the ASM, let be a directed multigraph with a global sink.
By *multigraph*, we mean that each edge of is assigned a nonnegative integer weight, and by *global* sink, we mean a vertex of out-degree such that every vertex of has a directed path into . We denote the vertices of by and the nonsink vertices by .

A *configuration* on is an element of , i.e., the assignment of a nonnegative integer to each nonsink vertex. We think of each integer as a number of grains of sand being placed at the corresponding vertex. A *divisor* on is an element of , i.e., an element in the free abelian group on all of the vertices. In the context of divisors, it is sometimes useful to think of assigning dollars to each vertex, with negative integers signifying a debt.

A configuration is *stable* at a vertex if
, and itself is stable if it is stable at each
nonsink vertex. If is unstable at , the vertex can be *fired*
(*toppled*) by removing grains of sand from and
adding grains of sand to the neighbors of sand, determined by the weights of
the edges leaving .

**Example.** Consider the graph:

All edges have weight except for the edge from vertex 1 to vertex 3, which have weight . If we let with the indicated number of grains of sand on vertices 1, 2, and 3, respectively, then only vertex 1, whose out-degree is 4, is unstable. Firing vertex 1 gives a new configuration . Here, grains have left vertex 1. One of these has gone to the sink vertex (and forgotten), one has gone to vertex 1, and two have gone to vertex 2, since the edge from 1 to 2 has weight 2. Vertex 3 in the new configuration is now unstable. The Sage code for this example looks like this:

```
Create the sandpile:
sage: load sandpile.sage
sage: g = {'sink':{},
1:{'sink':1, 2:1, 3:2},
2:{1:1, 3:1},
3:{1:1, 2:1}}
sage: S = Sandpile(g, 'sink')
sage: S.show(edge_labels=true) # to display the graph
Create the configuration:
sage: c = {1:5, 2:0, 3:1}
sage: S.out_degree()
{1: 4, 2: 2, 3: 2, 'sink': 0}
Fire vertex one:
sage: S.fire_vertex(1,c)
{1: 1, 2: 1, 3: 3}
sage: c
{1: 1, 2: 1, 3: 3}
Stabilize the rest of the way:
sage: S.stabilize(c)
[{1: 2, 2: 1, 3: 1}, {1: 1, 2: 2, 3: 3}]
```

Since vertex 3 has become unstable, it can be fired, which causes vertex 2 to become unstable, etc. Since there is a global sink, repeated firings eventually lead to a stable configuration. The last line of the Sage code, above, is a list, the first element of which is the resulting stable configuration, . (The second component records how many times each vertex fired in the stabilization.)

Since here is a global sink, every configuration will stabilize after a finite number of vertex-firings. It is not obvious, but the resulting stabilization is independent of the order in which unstable vertices are fired. Thus, each configuration stabilizes to a unique stable configuration.

Fix an order on the vertices of . The *Laplacian* of is

where is the diagonal matrix of out-degrees of the vertices and is the
adjacency matrix whose -th entry is the weight of the edge from vertex
to vertex , which we take to be if there is no edge. The *reduced
Laplacian*, , is the submatrix of the Laplacian formed by removing the row and
column corresponding to the sink vertex. Firing a vertex of a configuration is the same as subtracting the row of the reduced Laplacian.

**Example.** (Continued.)

```
sage: S.vertices() # here is the ordering of the vertices
[1, 2, 3, 'sink']
sage: S.laplacian()
[ 4 -1 -2 -1]
[-1 2 -1 0]
[-1 -1 2 0]
[ 0 0 0 0]
sage: S.reduced_laplacian()
[ 4 -1 -2]
[-1 2 -1]
[-1 -1 2]
The configuration we considered previously:
sage: c = S.list_to_config([5,0,1])
sage: c
{1: 5, 2: 0, 3: 1}
Firing vertex 1 is the same as substracting the
corresponding row from the reduced Laplacian:
sage: S.fire_vertex(1,c)
{1: 1, 2: 1, 3: 3}
sage: S.reduced_laplacian()[0]
(4, -1, -2)
sage: vector([5,0,1]) - vector([4,-1,-2])
(1, 1, 3)
```

Imagine an experiment in which grains of sand are dropped one-at-a-time onto a
graph, pausing to allow the configuration to stabilize between drops. Some
configurations will only be seen once in this process. For example, for most
graphs, once sand is dropped on the graph, no addition of sand+stabilization
will result in a graph empty of sand. Other configurations—the so-called
*recurrent configurations*—will be seen infinitely often as the process is
repeated indefinitely.

To be precise, a configuration is *recurrent* if (i) it is stable, and (ii)
given any configuration , there is a configuration such that
, the stabilization of .

The *maximal-stable* configuration, denoted is defined by
for all nonsink vertices . It is clear that is recurrent. Further, it is not hard to see that a configuration is recurrent if and only if it has the form for some configuration .

**Example.** (Continued.)

```
sage: S.recurrents(verbose=false)
[[3, 1, 1], [2, 1, 1], [3, 1, 0]]
sage: c = S.list_to_config([2,1,1])
sage: c
{1: 2, 2: 1, 3: 1}
sage: S.is_recurrent(c)
True
sage: S.max_stable()
{1: 3, 2: 1, 3: 1}
Adding any configuration to the max-stable configuration and stabilizing
yields a recurrent configuration.
sage: x = S.list_to_config([1,0,0])
sage: S.add(x,S.max_stable())
{1: 4, 2: 1, 3: 1}
sage: S.stabilize(_)
[{1: 3, 2: 1, 3: 0}, {1: 2, 2: 3, 3: 4}]
sage: c = _[0]
sage: c
{1: 3, 2: 1, 3: 0}
sage: S.is_recurrent(c)
True
```

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 is the burning configuration, is its
script, and is the reduced Laplacian, then .
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 with burning configuration having script :

- is recurrent;
- stabilizes to ;
- the firing vector for the stabilization of is .

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.

**Example.**

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

The collection of stable configurations forms a commutative monoid with addition defined as ordinary addition followed by stabilization. The identity element is the all-zero configuration. This monoid is a group exactly when the underlying graph is a DAG (directed acyclic graph).

The recurrent elements form a submonoid which turns out to be a group. This
group is called the *sandpile group* for , denoted . Its identity element is
usually not the all-zero configuration (again, only in the case that
is a DAG). So find the identity element is an interesting question.

Let and fix an ordering of the nonsink vertices. Let denote the column-span of , the tranpose of the reduced Laplacian. It is a theorem that

Thus, the number of element of the sandpile group is , which by the matrix-tree theorem is the number of weighted trees directed into the sink.

**Example.** (Continued.)

```
sage: S.group_order()
3
sage: S.elementary_divisors()
[1, 1, 3]
sage: S.reduced_laplacian().dense_matrix().smith_form()
([1 0 0]
[0 1 0]
[0 0 3],
[ 0 0 1]
[ 1 0 0]
[ 0 1 -1],
[3 1 4]
[4 1 6]
[4 1 5])
Adding the identity to any recurrent configuration and stabilizing yields
the same recurrent configuration:
sage: S.identity()
{1: 3, 2: 1, 3: 0}
sage: i = S.identity()
sage: m = S.max_stable()
sage: S.stably_add(i,m)[0] == m
True
```

The sandpile model was introduced by Bak, Tang, and Wiesenfeld in the paper,
*Self-organized criticality: an explanation of 1/ƒ noise* [BTW]. The term
*self-organized criticality* has no generally accepted definition, but can be
loosely taken to mean *like the sandpile model on a grid-graph*. The grid
graph is just a grid with an extra sink vertex. The vertices on the interior
of each side have one edge to the sink, and the corner vertices have an edge of
weight . Thus, every nonsink vertex has out-degree .

One property that is taken as characteristic of self-organized criticality is the presence of a power law (by which is meant an exponential law) describing the behavior of some ascpect of the system. For the grid graph, computer experiments—I do not think there is a proof, yet—indicate that the distribution of avalanche sizes obeys a power law. In the example below, the size of an avalanche is taken to be the sum of the number of imes each vertex fires.

**Example.**

```
Distribution of avalanche sizes:
sage: S = Sandpile(grid(10,10),'sink')
sage: S.set_config(S.max_stable())
sage: a = [sum(S.stabilize(S.add_random())[1].values()) for i in range(10000)]
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(['N','D(N)'])
sage: p
```

A reference for this section is *Riemann-Roch and Abel-Jacobi theory on a finite
graph* [BN].

A *divisor* on is an element of the free abelian group on its vertices,
including the sink. Suppose, as above, that the vertices of have
been ordered, and that is the column span of the transpose of the
Laplacian. A divisor is then identified with an element .
Two divisors are *linearly equivalent* if they differ by an element of
. A divisor is *effective*, written , if for
each , i.e., if . The *degree* of a divisor, ,
is . The divisors of degree zero modulo linear
equivalence form the *Picard group*, or *Jacobian* of the graph. For an
undirected graph, the Picard group is isomorphic to the sandpile group.

The *complete linear system* for a divisor , denoted , is the collection
of effective divisors linearly equivalent to To describe the Riemann-Roch
theorem in this context, suppose that is
an undirected, unweighted graph. The *dimension*, of
the linear system is if and otherwise is the greatest
integer such that for all effective divisors of degree .
Define the *canonical divisor* by and the *genus*
by . The
Riemann-Roch theorem says that for any divisor ,

**Example.** Some of the following calculations require the installation of *4ti2*.

```
The sandpile on the complete graph on 5 vertices:
sage: Gamma = graphs.CompleteGraph(5)
sage: S = Sandpile(Gamma,0)
The num_edges method counts each undirected edge twice:
sage: g = S.num_edges()/2 - S.num_verts() + 1
A divisor on the graph:
sage: D = S.list_to_div([1,2,2,0,2])
Verify the Riemann-Roch theorem:
sage: degD = sum(D.values())
sage: K = S.all_k_div(2)
sage: F = S.subtract(K,D)
sage: r_of_D = S.r_of_D(D)[0]
0
1
2
sage: r_of_F = S.r_of_D(F)[0]
0
sage: r_of_D - r_of_F == degD + 1 - g
True
The effective divisors linearly equivalent to D:
sage: S.effective_div(D)
[{0: 0, 1: 1, 2: 1, 3: 4, 4: 1},
{0: 4, 1: 0, 2: 0, 3: 3, 4: 0},
{0: 1, 1: 2, 2: 2, 3: 0, 4: 2}]
```

A reference for the following material is in the works [PP].

Let , and fix an ordering on the nonsink vertices of . let
denote the column-span of
, the tranpose of the reduced Laplacian. Label vertex with the
indeterminate , and let .
(Here, denotes the sink vertex of .) The *sandpile ideal* or
*toppling ideal* is the lattice ideal for :

where for .

For each define where and so that . Then, for each , define . It then turns out that

where is the -th standard basis vector and is any burning configuration.

The affine coordinate ring, is isomorphic to the group algebra of the sandpile group,

The standard term-ordering on is graded reverse lexigraphical order with if vertex is further from the sink than vertex . If is the script for a burning configuration (not necessarily minimal), then

is a Groebner basis for .

Now let , where
corresponds to the sink vertex. The *homogeneous sandpile ideal*, denoted
, is obtaining by homogenizing with respect to . Let be the
(full) Laplacian, and be the column span of
its transpose, Then is the lattice ideal for :

This ideal can be calculated by saturating the ideal

with respect to the product of the indeterminates: (extending the operator in the obvious way). A Groebner basis with respect to the degree lexicographic order describe above (with the smallest vertex), is obtained by homogenizing each element of the Groebner basis for the non-homogeneous sandpile ideal with respect to

**Example.**

```
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.ring()
// characteristic : 0
// number of vars : 6
// block 1 : ordering dp
// : names x_5 x_4 x_3 x_2 x_1 x_0
// block 2 : ordering C
The homogeneous sandpile ideal:
sage: S.ideal()
x_2-x_0,
x_3^2-x_5*x_0,
x_5*x_3-x_0^2,
x_4^2-x_3*x_1,
x_5^2-x_3*x_0,
x_1^3-x_4*x_3*x_0,
x_4*x_1^2-x_5*x_0^2
its resolution:
sage: S.resolution()
'R <-- R^7 <-- R^19 <-- R^25 <-- R^16 <-- R^4'
and Betti table:
sage: S.betti()
0 1 2 3 4 5
------------------------------------------
0: 1 1 - - - -
1: - 4 6 2 - -
2: - 2 7 7 2 -
3: - - 6 16 14 4
------------------------------------------
total: 1 7 19 25 16 4
The Hilbert function:
sage: S.hilbert_function()
[1, 5, 11, 15]
and its first differences (which counts the number of superstable
configurations in each degree):
sage: S.first_diffs_hilb()
[1, 4, 6, 4]
sage: x = [sum(i) for i in S.superstables(False)]
sage: sorted(x)
[0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]
The degree in which the Hilbert function equals the Hilbert polynomial, the
latter always being a constant in the case of a sandpile ideal:
sage: S.postulation()
3
```

The *zero set* for the sandpile ideal is

the set of simultaneous zeros of the polynomials in Letting denote the unit circle in the complex plane, is a finite subgroup of , isomorphic to the sandpile group. The zero set is actually linearly isomorphic to a faithful representation of the sandpile group on

**Example.** (Continued.)

```
sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0)
sage: S.ideal()
x_1^2-x_2^2,
x_1*x_2^3-x_0^4,
x_2^5-x_1*x_0^4
Approximation to the zero set (setting ``x_0 = 1``):
sage: S.solve()
[[0.707107*I - 0.707107, 0.707107 - 0.707107*I],
[-0.707107*I - 0.707107, 0.707107*I + 0.707107],
[-1*I, -1*I],
[I, I],
[0.707107*I + 0.707107, -0.707107*I - 0.707107],
[0.707107 - 0.707107*I, 0.707107*I - 0.707107],
[1, 1],
[-1, -1]]
sage: len(_) == S.group_order()
True
The zeros are generated as a group by a single vector:
sage: S.points()
[[e^(1/4*I*pi), e^(-3/4*I*pi)]]
```

The homogeneous sandpile ideal, , has a free resolution graded by the
divisors on modulo linear equivalence. (See the section on
*Discrete Riemann Surfaces* for the language of
divisors and linear equivalence.) Let
, as above, and let
denote the group of divisors modulo rational equivalence. Then
is graded by by letting for
each monomial . The minimal free resolution of has the form

where the are the *Betti numbers* for .

For each divisor class , define a simplicial complex,

The Betti number equals the dimension over of the -th reduced homology group of :

```
sage: S = Sandpile({0:{},1:{0: 1, 2: 1, 3: 4},2:{3: 5},3:{1: 1, 2: 1}},0)
Representatives of all divisor classes with nontrivial homology:
sage: p = S.betti_complexes()
sage: p[0]
[{0: -8, 1: 5, 2: 4, 3: 1},
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 2), (3,)}]
The homology associated with the first divisor in the list:
sage: D = p[0][0]
sage: S.effective_div(D)
[{0: 0, 1: 1, 2: 1, 3: 0}, {0: 0, 1: 0, 2: 0, 3: 2}]
sage: [S.support(E) for E in S.effective_div(D)]
[[1, 2], [3]]
sage: S.Dcomplex(D)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 2), (3,)}
sage: S.Dcomplex(D).homology()
{0: Z, 1: 0}
The minimal free resolution:
sage: S.resolution()
'R <-- 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
The degrees and ranks of the homology groups foreach element of the list p (compare with the
Betti table, above):
sage: [[sum(d[0].values()),d[1].betti()] for d in p]
[[2, {0: 1, 1: 0}],
[3, {0: 0, 1: 1, 2: 0}],
[2, {0: 1, 1: 0}],
[3, {0: 0, 1: 1, 2: 0}],
[2, {0: 1, 1: 0}],
[3, {0: 0, 1: 1, 2: 0}],
[2, {0: 1, 1: 0}],
[3, {0: 0, 1: 1}],
[2, {0: 1, 1: 0}],
[3, {0: 0, 1: 1, 2: 0}],
[5, {0: 0, 1: 0, 2: 1}]]
```

It is assumed that Sage is already installed. If not, please see
the main Sage homepage for installation instructions.
To use `sandpile.sage`:

- download sandpile.sage
- start Sage, and issue the command

`sage: load sandpile.sage`

You may need to give the full path name to `sandpile.sage`.

Warning

The methods for computing linear systems of divisors and their corresponding simplicial complexes require the installation of 4ti2.

To make 4ti2 usable from Sage Sandpiles:

- download the program from the 4ti2 homepage, and follow the installation instructions given there
- open
`sandpiles.sage`in your favorite text editor and edit the following line (near the beginning of the file, near the copyright statement and the start of the definition of the Sandpile class), replacing the`path_to_zsolve`string with the path to the executables in your 4ti2 directory:

```
# set the following if 4ti2 is installed
path_to_zsolve = '/home/davidp/math/sandpile/4ti2/linux_x86/'
```

- start Sage and load
`sandpiles.sage`as described above.

Most of `sandpile.sage` consists of the definition of the class
`Sandpile` representing the abelian sandpile model on a graph .
Initialization has the form

`sage: S = Sandpile(graph, sink)`

where `graph` represents a graph and `sink` is the key for the sink
vertex. There are four possible forms for `graph`:

- a Python dictionary of dictionaries:

```
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}}
```

Each key is the name of a vertex. Next to each vertex name is a dictionary
consisting of pairs: `vertex: weight`. Each pair represents a directed edge
emanating from and ending at `vertex` having (non-negative integer) weight
equal to `weight`. Loops are allowed. In the example above, all of the weights are 1.

- a Python dictionary of lists:

```
sage: g = {0: [], 1: [0, 3, 4], 2: [0, 3, 5],
3: [2, 5], 4: [1, 3], 5: [2, 3]}
```

This is a short-hand when all of the edge-weights are equal to 1. The above example is for the same displayed graph.

- a Sage graph (of type
`sage.graphs.graph.Graph`):

```
sage: g = graphs.CompleteGraph(5)
sage: S = Sandpile(g, 0)
sage: type(g)
<class 'sage.graphs.graph.Graph'>
```

- a Sage digraph:

```
sage: S = Sandpile(digraphs.RandomDirectedGNC(6), 0)
sage: S.show()
```

See http://sagemath.org/doc/reference/sage/graphs/graph_generators.html for more information on the Sage graph library and graph constructors.

Each of these four formats is preprocessed by the Sandpile class so that,
internally, the graph is represented by the dictionary of dictionaries format
first presented. This internal format is returned by `dict()`:

```
sage: S = Sandpile({0:[], 1:[0, 3, 4], 2:[0, 3, 5],
3: [2, 5], 4: [1, 3], 5: [2, 3]},0)
sage: S.dict()
{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}}
```

Note

The user is responsible for assuring that each vertex has a directed path into the designated sink. If the sink has out-edges, these will be ignored for the purposes of sandpile calculations (but not calculations on divisors).

**Code for checking whether a given vertex is a sink:**

```
sage: S = Sandpile({0:[], 1:[0, 3, 4], 2:[0, 3, 5],
3: [2, 5], 4: [1, 3], 5: [2, 3]},0)
sage: [S.distance(v,0) for v in S.vertices()] # 0 is a sink
[0, 1, 1, 2, 2, 2]
sage: [S.distance(v,1) for v in S.vertices()] # 1 is not a sink
[+Infinity, 0, +Infinity, +Infinity, 1, +Infinity]
```

Here is a summary of Sandpile methods. The summary is followed by a list of complete descriptions of these methods. There are many more methods available for a Sandpile, e.g., those inherited from the class DiGraph. To see them all, enter

`sage: dir(Sandpile)`

**Summary of methods.**

*add(c, d)*— The sum of`c`and`d`without stabilizing.*add_random(config)*— Add a random grain of sand to`config`.*all_k_config(k)*— The configuration having`k`grains of sand on each vertex.*all_k_div(k)*— The divisor having`k`grains of sand on each vertex.*betti(verbose)*— The Betti table for the homogeneous sandpile ideal.*betti_complexes()*— All simplicial complexes arising from divisors having nontrivial homology.*burning_config()*— A minimal burning configuration.*burning_script()*— A script for the minimal burning configuration.*compare_configs(c, d)*— Is`c`at least as large as`d`at each vertex?*config()*— The current saved configuration.*config_to_list(config)*— Convert a configuration into a list.*Dcomplex(D)*— The simplicial complex determined by the supports of effective divisors linearly equivalent to`D`.*dict()*— Returns a dictionary of dictionaries representing a directed graph.*div_to_list(div)*— Convert a divisor into a list in the order determined by`vertices()`.*dualize(config)*— Returns`max_stable - config`.*effective_div(div)*— Returns all effective divisors linearly equivalent to .*element_order(config)*— Returns the order of the recurrent element equivalent to`config`.*elementary_divisors()*— The elementary divisors of the sandpile group.*equivalent_recurrent(config)*— The recurrent configuration equivalent to the given configuration (+ firing vector).*equivalent_superstable(config)*— The superstable configuration equivalent to the given configuration (+ firing vector).*fire_vertex(v, config)*— Fire (topple) a given vertex of a configuration.*first_diffs_hilb()*— The first differences of the Hilbert function of the homogeneous sandpile ideal.*groebner()*— A Groebner basis for the homogeneous sandpile ideal with respect to the standard sandpile ordering (see`ring`).*group_order()*— The size of the sandpile group.*hilbert_function()*— The Hilbert function of the homogeneous sandpile ideal.*ideal()*— The saturated, homogeneous sandpile ideal.*identity()*— The identity configuration.*in_degree(v)*— The in-degree of a vertex or a list of all in-degrees.*is_recurrent(config)*Is`config`is recurrent, i.e., an element of the sandpile group?*is_stable(config)*— Is`config`stable?*is_superstable(config)*— Is`config`superstable, i.e., does it not permit multiset firings?*is_symmetric(c, orbits)*— Is is constant over the vertices in each sublist of`orbits`?*laplacian()*— The Laplacian matrix of the graph.*linear_system(div)*— The complete linear system of a divisor.*list_to_config(L)*— Convert a list into a configuration.*list_to_div(L)*— Convert a list of integers into a divisor.*max_stable()*— The maximal stable configuration.*nonsink_vertices()*— The names of the nonsink vertices.*out_degree(v)*— The out-degree of a vertex or a list of all out-degrees.*points()*— Generators for the multiplicative group of zeros of the sandpile ideal.*postulation()*— The postulation number of the sandpile ideal.*r_of_D(D)*— Computes`r(D)`and an effective divisor`F`such that`|D - F|`is empty.*recurrent_difference(config1, config2)*— A recurrent configuration equivalent to the`config1 - config2`.*recurrents(verbose)*— The list of recurrent configurations.*reduced_laplacian()*— The reduced Laplacian matrix of the graph.*reset_config(config)*— Turn`config`into the zero configuration.*resolution(verbose)*— The minimal free resolution of the homogeneous sandpile ideal.*ring()*— The ring containing the homogeneous sandpile ideal.*set_config(config)*— Sets the current configuration to`config`.*sink()*— The identifier for the sink vertex.*solve()*— Computes approximations of the complex affine zeros of the sandpile ideal.*stabilize(config)*— The stabilized configuration and its firing vector.*stabilize_one_step(config)*— Fire each unstable vertex in`config`(+ firing vector).*stably_add(config1, config2)*— The stabilization of the sum of two configurations (+ firing vector).*subtract(c, d)*— The difference of`c - d`of configurations or divisors.*superstables(verbose)*— The superstable configurations (-parking functions).*support(D)*— The list of keys of nonzero values of the dictionary`D`.*symmetric_recurrents(orbits)*— The list of symmetric recurrent configurations.*unsaturated_ideal()*— The unsaturated, homogeneous sandpile ideal.*unstable(config)*— The list of unstable vertices in`config`.*version()*— The version number of Sage Sandpiles.*vertices()*— A list of the vertices.

**Complete descriptions of methods.**

**add(c, d)**

Returns the sum of

canddwithout stabilizing.INPUT:

c,d- dict (configurations or divisors)OUTPUT:

dict (configuration or divisor)

EXAMPLES:

Adding configurations:

sage: S = sandlib('generic') sage: m = S.max_stable() sage: m {1: 2, 2: 2, 3: 1, 4: 1, 5: 1} sage: S.add(m, m) {1: 4, 2: 4, 3: 2, 4: 2, 5: 2}Adding divisors:

sage: D = S.list_to_div([1,-3,2,0,4,-2]) sage: E = S.list_to_div([2, 3,4,1,1,-2]) sage: S.add(D,E) {0: 3, 1: 0, 2: 6, 3: 1, 4: 5, 5: -4}

—

**add_random(config=None)**

Add one grain of sand to a random nonsink vertex. The

configargument can be a configuration or a divisor.INPUT:

config(optional) - dict (a configuration or divisor)OUTPUT:

None

EXAMPLES:

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: S = Sandpile(grid(4,4),'sink') sage: a = [sum(S.stabilize(S.add_random())[1].values()) for i in range(1000)] sage: b = [[log(i+1),log(a.count(i))] for i in [0..max(a)] if a.count(i)] sage: list_plot(b)Also works for divisors:

sage: S = sandlib('generic') sage: D = S.list_to_div([0,0,0,0,0,0]) sage: S.add_random(D) sage: D #random {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 1}

—

**all_k_config(k)**

The configuration having

kgrains of sand on each vertex.INPUT:

k- integerOUTPUT:

dict (configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: S.all_k_config(7) {1: 7, 2: 7, 3: 7, 4: 7, 5: 7}

—

**all_k_div(k)**

The divisor having

kgrains of sand on each vertex.INPUT:

k- integerOUTPUT:

dict (divisor)

EXAMPLES:

sage: S = sandlib('generic') sage: S.all_k_div(7) {0: 7, 1: 7, 2: 7, 3: 7, 4: 7, 5: 7}

—

**betti(verbose=True)**

Computes the Betti table for the homogeneous sandpile ideal. If

verboseisTrue, it prints the standard Betti table, otherwise, it returns a less formated table.INPUT:

verbose(optional) - booleanOUTPUT:

Betti numbers for the sandpile

EXAMPLE:

sage: S = sandlib('generic') sage: S.betti() 0 1 2 3 4 5 ------------------------------------------ 0: 1 1 - - - - 1: - 4 6 2 - - 2: - 2 7 7 2 - 3: - - 6 16 14 4 ------------------------------------------ total: 1 7 19 25 16 4

—

**betti_complexes()**

Returns a list of all the divisors with nonempty linear systems whose corresponding simplicial complexes have nonzero homology in some dimension. Each such divisors is returned with its corresponding simplicial complex.

INPUT:

None

OUTPUT:

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

EXAMPLES:

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 (0, 1, 2, 3) and facets {(1, 2), (3,)}] sage: S.resolution() 'R <-- 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

—

**burning_config()**

A minimal burning configuration.

INPUT:

None

OUTPUT:

dict (configuration)

EXAMPLES:

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

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 configurationis 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 correspondingburning scriptgives the integer-linear combination needed to obtain the burning configuration. So if is the burning configuration, is its script, and is the reduced Laplacian, then . Theminimal burning configurationis 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 with burning configuration having script :

- is recurrent;
- stabilizes to ;
- the firing vector for the stabilization of is .

—

**burning_script()**

A script for the minimal burning configuration.

INPUT:

None

OUTPUT:

dict

EXAMPLES:

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

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 configurationis 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 correspondingburning scriptgives the integer-linear combination needed to obtain the burning configuration. So if is the burning configuration, is its script, and is the reduced Laplacian, then . Theminimal burning configurationis 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 with burning configuration having script :

- is recurrent;
- stabilizes to ;
- the firing vector for the stabilization of is .

—

**compare_configs(c, d)**

Returns True if each

cis at least as large asdat each vertex.INPUT:

c,d- dict (configurations or divisors)OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: c = S.all_k_config(5) sage: d = S.all_k_config(4) sage: S.compare_configs(c,d) True sage: S.compare_configs(d,c) False

—

**config()**

The current saved configuration.

INPUT:

None

OUTPUT:

None

EXAMPLES:

sage: G = sandlib('generic') sage: G.set_config(G.max_stable()) sage: G.config() == G.max_stable() True sage: G.add(G.max_stable(),G.identity()) {1: 4, 2: 4, 3: 2, 4: 2, 5: 1} sage: G.config() {1: 4, 2: 4, 3: 2, 4: 2, 5: 1}NOTES:

Each sandpile has a saved configuration (initially the zero configuration) The configuration can be set with the

set_config()method and may be affected by certain other methods, e.g.,stabilize(),add(), andadd_random()—usually those methods that output configurations.

—

**config_to_list(config)**

Convert a configuration into a list in the order determined by

nonsink_vertices().INPUT:

config- dict (configuration)OUTPUT:

list of integers

EXAMPLES:

sage: S = sandlib('generic') sage: c = S.list_to_config([5,4,3,2,1]) sage: c {1: 5, 2: 4, 3: 3, 4: 2, 5: 1} sage: S.config_to_list(c) [5, 4, 3, 2, 1]NOTES:

The method checks the length of

configbut not the types of its entries.

—

**Dcomplex(D)**

Returns the simplicial complex determined by the supports of the effective divisors linearly equivalent to

D.INPUT:

D- dict (divisor)OUTPUT:

simplicial complex

EXAMPLES:

sage: S = sandlib('generic') sage: p = S.Dcomplex(S.list_to_div([0,1,2,0,0,1])) sage: p.homology() {0: 0, 1: Z x Z, 2: 0, 3: 0} sage: p.f_vector() [1, 6, 15, 9, 1] sage: p.betti() {0: 0, 1: 2, 2: 0, 3: 0}

—

**dict()**

Returns a dictionary of dictionaries representing a directed graph.

INPUT:

None

OUTPUT:

dict

EXAMPLES:

sage: G = sandlib('generic') sage: G.dict() {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: G.sink() 0

—

**div_to_list(div)**

Convert a divisor into a list in the order determined by

vertices().INPUT:

div- dict (divisor)OUTPUT:

list of integers

EXAMPLES:

sage: S = sandlib('generic') sage: c = S.list_to_div([6,5,4,3,2,1]) sage: c {0: 6, 1: 5, 2: 4, 3: 3, 4: 2, 5: 1} sage: S.div_to_list(c) [6, 5, 4, 3, 2, 1]NOTES:

The method checks the length of

divbut not the types of its entries.

—

**dualize(config=None)**

Returns

max_stable - config.INPUT:

config(optional) - dict (configuration)OUTPUT:

dict (configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: S.dualize(S.max_stable()) {1: 0, 2: 0, 3: 0, 4: 0, 5: 0}

—

**effective_div(div):**

Returns all effective divisors linearly equivalent to

div.INPUT:

div- dict (divisor)OUTPUT:

list (of divisors)

EXAMPLES:

sage: S = sandlib('generic') sage: D = S.list_to_div([0,0,0,0,0,2]) sage: S.effective_div(D) [{0: 1, 1: 0, 2: 0, 3: 1, 4: 0, 5: 0}, {0: 0, 1: 0, 2: 1, 3: 1, 4: 0, 5: 0}, {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 2}] sage: [S.div_to_list(d) for d in _] [[1, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 2]]

—

**element_order(config=None)**

Returns the order of the recurrent element equivalent to

config.INPUT:

config (optional) - configuration

OUTPUT:

integer

EXAMPLES:

sage: S = sandlib('generic') sage: [S.element_order(r) for r in S.recurrents()] [1, 3, 15, 5, 5, 15, 5, 15, 3, 15, 15, 5, 15, 15, 15]

—

**elementary_divisors()**

The elementary divisors of the sandpile group (a finite abelian group).

INPUT:

None

OUTPUT:

list of integers

EXAMPLES:

sage: S = sandlib('generic') sage: S.elementary_divisors() [1, 1, 1, 1, 15]

—

**equivalent_recurrent(config)**

Returns the recurrent configuration equivalent to the given configuration and also returns the corresponding firing vector.

INPUT:

config- dict (configuration)OUTPUT:

[configuration, firing_vector]EXAMPLES:

sage: S = sandlib('generic') sage: c = [0,0,0,0,0] sage: config = S.list_to_config([0,0,0,0,0]) sage: x = S.equivalent_recurrent(config) sage: x[0] == S.identity() True sage: r = [x[0][v] for v in S.nonsink_vertices()] sage: f = [x[1][v] for v in S.nonsink_vertices()] sage: vector(r) == vector(c) - vector(f)*S.reduced_laplacian() TrueNOTES:

Let be the reduced laplacian, the initial configuration, the returned configuration, and the firing vector. Then .

—

**equivalent_superstable(config)**

Returns the superstable configuration equivalent to the given configuration and also returns the corresponding firing vector.

INPUT:

config- dict (configuration)OUTPUT:

[configuration, firing_vector]EXAMPLES:

sage: S = sandlib('generic') sage: m = S.max_stable() sage: x = S.equivalent_superstable(m) sage: S.is_superstable(x[0]) True sage: s = [x[0][v] for v in S.nonsink_vertices()] sage: f = [x[1][v] for v in S.nonsink_vertices()] sage: m = [m[v] for v in S.nonsink_vertices()] sage: vector(s) == vector(m) - vector(f)*S.reduced_laplacian()NOTES:

Let be the reduced laplacian, the initial configuration, the returned configuration, and the firing vector. Then .

—

**fire_vertex(v, config=None)**

Fire (topple) a given vertex of a configuration.

INPUT:

v- vertex nameconfig- dictOUTPUT:

dict (configuration)

EXAMPLES:

sage: G = sandlib('generic') sage: c = G.add(G.max_stable(),G.identity()) sage: G.unstable(c) [1, 2, 3, 4] sage: c {1: 4, 2: 4, 3: 2, 4: 2, 5: 1} sage: G.fire_vertex(1,c) {1: 1, 2: 4, 3: 3, 4: 3, 5: 1}NOTES:

This method fires vertex

vinconfigprovidedvis a nonsink, unstable vertex. Returns the result (and modifiesself.config). The vertexvis fired only once, even if the result leavesvunstable.

—

**first_diffs_hilb()**

Returns the first differences of the Hilbert function of the homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

list of nonnegative integers

EXAMPLES:

sage: S = sandlib('generic') sage: S.hilbert_function() [1, 5, 11, 15] sage: S.first_diffs_hilb() [1, 4, 6, 4]

—

**groebner()**

Returns a Groebner basis for the homogeneous sandpile ideal with respect to the standard sandpile ordering (see

ring).INPUT:

None

OUTPUT:

Groebner basis

EXAMPLES:

sage: S = sandlib('generic') sage: S.groebner() x_2-x_0, x_3^2-x_5*x_0, x_5*x_3-x_0^2, x_4^2-x_3*x_1, x_5^2-x_3*x_0, x_1^3-x_4*x_3*x_0, x_4*x_1^2-x_5*x_0^2

—

**group_order()**

Returns the size of the sandpile group.

INPUT:

None

OUTPUT:

int

EXAMPLES:

sage: G = sandlib('generic') sage: G.group_order() 15

—

**hilbert_function()**

Returns the Hilbert function of the homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

list of nonnegative integers

EXAMPLES:

sage: S = sandlib('generic') sage: S.hilbert_function() [1, 5, 11, 15]

—

**ideal()**

The saturated, homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

ideal

EXAMPLES:

sage: S = sandlib('generic') sage: S.ideal() x_2-x_0, x_3^2-x_5*x_0, x_5*x_3-x_0^2, x_4^2-x_3*x_1, x_5^2-x_3*x_0, x_1^3-x_4*x_3*x_0, x_4*x_1^2-x_5*x_0^2

—

**identity()**

Returns the identity configuration.

INPUT:

None

OUTPUT:

dict (the identity configuration)

EXAMPLES:

sage: G = sandlib('generic') sage: e = G.identity() sage: x = G.stably_add(e, G.max_stable()) sage: x [{1: 2, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 1, 2: 5, 3: 7, 4: 1, 5: 6}] sage: G.config_to_list(x[0]) [2, 2, 1, 1, 1] sage: G.config_to_list(G.max_stable()) [2, 2, 1, 1, 1]

—

**in_degree(v=None)**

Return the in-degree of a vertex or a list of all in-degrees.

INPUT:

v- vertex name or NoneOUTPUT:

integer or list of integers

EXAMPLES:

sage: G = sandlib('generic') sage: G.in_degree(2) 2 sage: G.in_degree() {0: 2, 1: 1, 2: 2, 3: 4, 4: 1, 5: 2}

—

**is_recurrent(config=None)**

Returns True if

configis recurrent, i.e., is an element of the sandpile group.INPUT:

config(optional) - dict (configuration)OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.is_recurrent(S.identity()) True sage: c = S.list_to_config([0,0,0,0,0]) sage: S.is_recurrent(c) False

—

**is_stable(config=None)**

Returns True if

configis stable.INPUT:

config(optional) - dict (configuration)OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.is_stable(S.max_stable()) True sage: c = S.add(S.max_stable(), S.max_stable()) sage: S.is_stable(c) False

—

**is_superstable(config=None)**

Returns True if

configis superstable, i.e., it does not permit multiset firings.INPUT:

config(optional) - dict (configuration)OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.is_superstable(S.identity()) False sage: c = S.subtract(S.max_stable(), S.identity()) sage: S.is_superstable(c) True

—

**is_symmetric(c, orbits)**

This function checks if is constant over the vertices in each sublist of

orbits.INPUT:

c- configurationorbit- list of lists of verticesOUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('kite') sage: S.dict() {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 = S.list_to_config([1, 2, 2, 3]) sage: S.is_symmetric(c,[[2,3]]) True

—

**laplacian()**

Returns the Laplacian matrix of the graph.

INPUT:

None

OUTPUT:

matrix

EXAMPLES:

sage: G = sandlib('generic') sage: G.laplacian() [ 0 0 0 0 0 0] [-1 3 0 -1 -1 0] [-1 0 3 -1 0 -1] [ 0 0 -1 2 0 -1] [ 0 -1 0 -1 2 0] [ 0 0 -1 -1 0 2]

—

**linear_system(div)**

Returns the complete linear system of a divisor.

INPUT:

div- divisorOUTPUT:

dict -

{num_homog: int, homog:list, num_inhomog:int, inhomog:list}EXAMPLES:

sage: S = sandlib('generic') sage: D = S.list_to_div([0,0,0,0,0,2]) sage: S.linear_system(D) {'homog': [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0]], 'inhomog': [[0, 0, -1, -1, 0, -2], [0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0]], 'num_homog': 2, 'num_inhomog': 3}NOTES:

If is the Laplacian, an arbitrary such that has the form where is in

inhomgand is in the integer span ofhomogin the output oflinear_system(D).WARNING:

This method requires 4ti2. After local installation of 4ti2, set the

path_to_zsolveat the beginning ofsandpile.sage.

—

**list_to_config(L)**

Convert a list into a configuration.

INPUT:

L- list of nonnegative integersOUTPUT:

dict (configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: S.list_to_config([3,6,4,2,4]) {1: 3, 2: 6, 3: 4, 4: 2, 5: 4}NOTES:

The method checks the length of

Lbut not the types of its entries.

—

**list_to_div(L)**

Convert a list of integers into a divisor.

INPUT:

L- list of integersOUTPUT:

dict (configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: S.list_to_div([6,5,4,3,2,1]) {0: 6, 1: 5, 2: 4, 3: 3, 4: 2, 5: 1}NOTES:

The method checks the length of

Lbut not the types of its entries.

—

**max_stable()**

Returns the maximal stable configuration.

INPUT:

None

OUTPUT:

dict (the maximal stable configuration)

EXAMPLES:

sage: G = sandlib('generic') sage: G.max_stable() {1: 2, 2: 2, 3: 1, 4: 1, 5: 1}

—

**nonsink_vertices()**

The names of the nonsink vertices.

INPUT:

None

OUTPUT:

None

EXAMPLES:

sage: G = sandlib('generic') sage: G.nonsink_vertices() [1, 2, 3, 4, 5]

—

**out_degree(v=None)**

Return the out-degree of a vertex or a list of all out-degrees.

INPUT:

v(optional) - vertex nameOUTPUT:

integer or list of integers

EXAMPLES:

sage: G = sandlib('generic') sage: G.out_degree(2) 3 sage: G.out_degree() {0: 0, 1: 3, 2: 3, 3: 2, 4: 2, 5: 2}

—

**points()**

Returns generators for the multiplicative group of zeros of the sandpile ideal.

INPUT:

None

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: S = sandlib('generic') sage: S.points() [[e^(4/5*I*pi), 1, e^(2/3*I*pi), e^(-34/15*I*pi), e^(-2/3*I*pi)]]

—

**postulation()**

Returns the postulation number of the sandpile ideal. This is the largest weight of a superstable configuration of the graph.

INPUT:

None

OUTPUT:

nonnegative integer

EXAMPLES:

sage: S = sandlib('generic') sage: S.postulation() 3

—

**r_of_D(D)**

Computes

r(D)and an effective divisorFsuch that|D - F|is empty.INPUT:

D- divisorOUTPUT:

tuple ((integer

r(D), divisorF))EXAMPLES:

sage: S = sandlib('generic') sage: D = S.list_to_div([0,0,0,0,0,4]) sage: E = S.r_of_D(D) 0 1 sage: E (1, {0: 0, 1: 1, 2: 0, 3: 1, 4: 0, 5: 0}) sage: F = E[1] sage: S.div_to_list(S.subtract(D,F)) [0, -1, 0, -1, 0, 4] sage: S.effective_div(S.subtract(D,F)) [] sage: S.r_of_D(S.list_to_div([0,0,0,0,0,-4])) (-1, {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: -4})

—

**recurrent_difference(config1, config2)**

Returns a recurrent configuration equivalent to the difference of the two given configurations,

config1 - config2.INPUT:

config1,config2- dict (configurations)OUTPUT:

dict (configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: c = S.recurrent_difference(S.max_stable(), S.max_stable()) sage: c {1: 2, 2: 2, 3: 1, 4: 1, 5: 0} sage: c == S.identity() True

—

**recurrents(verbose=True)**

Returns the list of recurrent configurations as dictionaries if

verboseisTrue, otherwise as lists of integers.INPUT:

verbose(optional) – booleanOUTPUT:

list (of recurrent configurations)

EXAMPLES:

sage: S = sandlib('generic') sage: S.recurrents() [{1: 2, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 2, 2: 2, 3: 0, 4: 1, 5: 1}, {1: 0, 2: 2, 3: 1, 4: 1, 5: 0}, {1: 0, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 1, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 1, 2: 2, 3: 0, 4: 1, 5: 1}, {1: 2, 2: 2, 3: 1, 4: 0, 5: 1}, {1: 2, 2: 2, 3: 0, 4: 0, 5: 1}, {1: 2, 2: 2, 3: 1, 4: 0, 5: 0}, {1: 1, 2: 2, 3: 1, 4: 1, 5: 0}, {1: 1, 2: 2, 3: 1, 4: 0, 5: 0}, {1: 1, 2: 2, 3: 1, 4: 0, 5: 1}, {1: 0, 2: 2, 3: 0, 4: 1, 5: 1}, {1: 2, 2: 2, 3: 1, 4: 1, 5: 0}, {1: 1, 2: 2, 3: 0, 4: 0, 5: 1}] sage: S.recurrents(false) [[2, 2, 1, 1, 1], [2, 2, 0, 1, 1], [0, 2, 1, 1, 0], [0, 2, 1, 1, 1], [1, 2, 1, 1, 1], [1, 2, 0, 1, 1], [2, 2, 1, 0, 1], [2, 2, 0, 0, 1], [2, 2, 1, 0, 0], [1, 2, 1, 1, 0], [1, 2, 1, 0, 0], [1, 2, 1, 0, 1], [0, 2, 0, 1, 1], [2, 2, 1, 1, 0], [1, 2, 0, 0, 1]]

—

**reduced_laplacian()**

Returns the reduced Laplacian matrix of the graph.

INPUT:

None

OUTPUT:

matrix

EXAMPLES:

sage: G = sandlib('generic') sage: G.laplacian() [ 0 0 0 0 0 0] [-1 3 0 -1 -1 0] [-1 0 3 -1 0 -1] [ 0 0 -1 2 0 -1] [ 0 -1 0 -1 2 0] [ 0 0 -1 -1 0 2] sage: G.reduced_laplacian() [ 3 0 -1 -1 0] [ 0 3 -1 0 -1] [ 0 -1 2 0 -1] [-1 0 -1 2 0] [ 0 -1 -1 0 2]NOTES:

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

—

**reset_config(config=None)**

Turn

configinto the zero configuration.INPUT:

config(optional) - dictOUTPUT:

None

EXAMPLES:

sage: G = sandlib('generic') sage: G.add(G.max_stable(),G.max_stable()) {1: 4, 2: 4, 3: 2, 4: 2, 5: 2} sage: G.config() {1: 4, 2: 4, 3: 2, 4: 2, 5: 2} sage: G.reset_config() sage: G.config() {1: 0, 2: 0, 3: 0, 4: 0, 5: 0}

—

**resolution(verbose=False)**

This function computes a minimal free resolution of the homogeneous sandpile ideal. If

verboseisTrue, then all of the mappings are returned. Otherwise, the resolution is summarized.INPUT:

verbose(optional) - booleanOUTPUT:

free resolution of the sandpile ideal

EXAMPLES:

sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0) sage: S.resolution() 'R <-- R^2 <-- R^1' sage: S.resolution(verbose=True) [1]: _[1]=x_1^2-x_2^2 _[2]=x_1*x_2^3-x_0^4 [2]: _[1]=x_1*x_2^3*gen(1)-x_0^4*gen(1)-x_1^2*gen(2)+x_2^2*gen(2) [3]: _[1]=0

—

**ring()**

The ring containing the homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

ring

EXAMPLES:

sage: S = sandlib('generic') sage: S.ring() // characteristic : 0 // number of vars : 6 // block 1 : ordering dp // : names x_5 x_4 x_3 x_2 x_1 x_0 // block 2 : ordering CNOTES:

The indeterminate corresponds to the -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).

—

**set_config(config)**

Sets the current configuration to

config.INPUT:

config- dictOUTPUT:

None

EXAMPLES:

sage: G = sandlib('generic') sage: G.set_config(G.identity()) sage: G.config() == G.identity() True

—

**sink()**

Returns the identifier for the sink vertex.

INPUT:

None

OUTPUT: Object (name for the sink vertex)

EXAMPLES:

sage: G = sandlib('generic') sage: G.sink() 0 sage: H = Sandpile(grid(2,2),'sink') sage: H.sink() 'sink' sage: type(H.sink()) <type 'str'>

—

**solve()**

Computes approximations of the complex affine zeros of the sandpile ideal.

INPUT:

None

OUTPUT:

list of complex numbers

EXAMPLES:

sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0) sage: S.solve() [[0.707107*I - 0.707107, 0.707107 - 0.707107*I], [-0.707107*I - 0.707107, 0.707107*I + 0.707107], [-1*I, -1*I], [I, I], [0.707107*I + 0.707107, -0.707107*I - 0.707107], [0.707107 - 0.707107*I, 0.707107*I - 0.707107], [1, 1], [-1, -1]] sage: len(_) 8 sage: S.group_order() 8NOTES:

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

points().

—

**stabilize(config=None)**

Returns the stabilized configuration and its firing vector.

INPUT:

config(optional) - dictOUTPUT:

list -

[out_config, firing_vector]EXAMPLES:

sage: S = sandlib('generic') sage: c = S.add(S.max_stable(), S.identity()) sage: S.stabilize(c) [{1: 2, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 1, 2: 5, 3: 7, 4: 1, 5: 6}]The Sandpiles current configuration is modified:

sage: S.config() {1: 2, 2: 2, 3: 1, 4: 1, 5: 1}

—

**stabilize_one_step(config=None)**

Fire each unstable vertex in

config, returning a list[out_config, firing_vector]whereout_configis the modified configuration.INPUT:

config(optional) - dictOUTPUT:

list of the form [dict, dict]

EXAMPLES:

sage: G = sandlib('generic') sage: c = G.add(G.max_stable(),G.identity()) sage: c {1: 4, 2: 4, 3: 2, 4: 2, 5: 1} sage: G.unstable(c) [1, 2, 3, 4] sage: G.stabilize_one_step(c) [{1: 2, 2: 2, 3: 3, 4: 1, 5: 3}, {1: 1, 2: 1, 3: 1, 4: 1, 5: 0}]

—

**stably_add(config1, config2)**

Returns the stabilization and corresponding firing vector of the sum of two configurations.

INPUT:

config1,config2- dict (configurations)OUTPUT:

list (

[stabilized sum, firing_vector])EXAMPLES:

sage: S = sandlib('generic') sage: S.stably_add(S.max_stable(), S.max_stable()) [{1: 2, 2: 2, 3: 0, 4: 1, 5: 1}, {1: 1, 2: 7, 3: 10, 4: 1, 5: 9}]

—

**subtract(c, d)**

Returns the difference,

c - d.INPUT:

c,d- dict (configurations or divisors)OUTPUT:

dict (configuration or divisor)

EXAMPLES:

Subtracting configurations:

sage: S = sandlib('generic') sage: m = S.max_stable() sage: m {1: 2, 2: 2, 3: 1, 4: 1, 5: 1} sage: S.subtract(m, m) {1: 0, 2: 0, 3: 0, 4: 0, 5: 0}Subtracting divisors:

sage: D = S.list_to_div([1,-3,2,0,4,-2]) sage: E = S.list_to_div([2, 3,4,1,1,-2]) sage: S.subtract(D,E) {0: -1, 1: -6, 2: -2, 3: -1, 4: 3, 5: 0}

—

**superstables(verbose=True)**

Returns the list of superstable configurations as dictionaries if

verboseisTrue, otherwise as lists of integers. The superstables are also known as -parking functions.INPUT:

verbose- booleanOUTPUT:

list (of superstable elements)

EXAMPLES:

sage: S = sandlib('generic') sage: S.superstables() [{1: 0, 2: 0, 3: 0, 4: 0, 5: 0}, {1: 0, 2: 0, 3: 1, 4: 0, 5: 0}, {1: 2, 2: 0, 3: 0, 4: 0, 5: 1}, {1: 2, 2: 0, 3: 0, 4: 0, 5: 0}, {1: 1, 2: 0, 3: 0, 4: 0, 5: 0}, {1: 1, 2: 0, 3: 1, 4: 0, 5: 0}, {1: 0, 2: 0, 3: 0, 4: 1, 5: 0}, {1: 0, 2: 0, 3: 1, 4: 1, 5: 0}, {1: 0, 2: 0, 3: 0, 4: 1, 5: 1}, {1: 1, 2: 0, 3: 0, 4: 0, 5: 1}, {1: 1, 2: 0, 3: 0, 4: 1, 5: 1}, {1: 1, 2: 0, 3: 0, 4: 1, 5: 0}, {1: 2, 2: 0, 3: 1, 4: 0, 5: 0}, {1: 0, 2: 0, 3: 0, 4: 0, 5: 1}, {1: 1, 2: 0, 3: 1, 4: 1, 5: 0}] sage: S.superstables(false) [[0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [2, 0, 0, 0, 1], [2, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 1, 1, 0], [0, 0, 0, 1, 1], [1, 0, 0, 0, 1], [1, 0, 0, 1, 1], [1, 0, 0, 1, 0], [2, 0, 1, 0, 0], [0, 0, 0, 0, 1], [1, 0, 1, 1, 0]]

—

**support(D)**

The input is a dictionary of integers. The output is a list of keys of nonzero values of the dictionary.

INPUT:

D- dict (configuration or divisor)OUTPUT:

list - support of the configuration or divisor

EXAMPLES:

sage: S = sandlib('generic') sage: c = S.identity() sage: S.config_to_list(c) [2, 2, 1, 1, 0] sage: S.support(c) [1, 2, 3, 4] sage: S.vertices() [0, 1, 2, 3, 4, 5]

—

**symmetric_recurrents(orbits)**

Returns the list of symmetric recurrent configurations.

INPUT:

orbits- list of lists partitioning the verticesOUTPUT:

list of recurrent configurations

EXAMPLES:

sage: S = sandlib('kite') sage: S.dict() {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}]NOTES:

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

—

**unsaturated_ideal()**

The unsaturated, homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

ideal

EXAMPLES:

sage: S = sandlib('generic') sage: S.unsaturated_ideal() x_1^3-x_4*x_3*x_0, x_2^3-x_5*x_3*x_0, x_3^2-x_5*x_2, x_4^2-x_3*x_1, x_5^2-x_3*x_2 sage: S.ideal() x_2-x_0, x_3^2-x_5*x_0, x_5*x_3-x_0^2, x_4^2-x_3*x_1, x_5^2-x_3*x_0, x_1^3-x_4*x_3*x_0, x_4*x_1^2-x_5*x_0^2

—

**unstable(config=None)**

The list of unstable vertices in

config.INPUT:

config(optional) - dictOUTPUT:

list of vertex names

EXAMPLES:

sage: G = sandlib('generic') sage: c = G.add(G.max_stable(),G.identity()) sage: G.unstable(c) [1, 2, 3, 4] sage: G.stabilize(c); sage: G.unstable(c) []

—

**version()**

Returns the version number of Sage Sandpiles.

INPUT:

None

OUTPUT:

string

EXAMPLES:

sage: S = sandlib('generic') sage: S.version() Sage Sandpiles Version 1.5

—

**vertices(boundary_first=False))**

Return a list of the vertices.

INPUT:

boundary_first- Return the boundary vertices first.EXAMPLE:

sage: P = graphs.PetersenGraph() sage: P.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]Note that the output of the vertices() function is always sorted. This is sub-optimal, speedwise, but note the following optimizations:

sage: timeit V = P.vertices() # not tested 100000 loops, best of 3: 8.85 [micro]s per loop sage: timeit V = list(P.vertex_iterator()) # not tested 100000 loops, best of 3: 5.74 [micro]s per loop sage: timeit V = list(P._nxg.adj.iterkeys()) # not tested 100000 loops, best of 3: 3.45 [micro]s per loopIn other words, if you want a fast vertex iterator, call the dictionary directly.

The following methods create Sandpiles. (See the examples in the complete descriptions.)

*aztec(n)*— The aztec diamond graph.*sandlib(selector)*— A collection of sandpiles.*grid(m, n)*— The grid graph.*random_graph(num_verts, p, directed, weight_max)*— A random graph.*random_DAG(num_verts, p, weight_max)*— A random directed acyclic graph.*glue_graphs(g, h, glue_g, glue_h)*— Glue two sandpiles together.

**Complete descriptions of methods.**

**aztec(n)**

The aztec diamond graph.

INPUT:

n - integer

OUTPUT:

dictionary for the aztec diamond graph

EXAMPLES:

sage: aztec(2) {(-3/2, -1/2): {}, (-3/2, 1/2): {}, (-1/2, -3/2): {'sink': 2, (-1/2, -1/2): 1, (1/2, -3/2): 1}, (-1/2, -1/2): {(-3/2, -1/2): 1, (-1/2, -3/2): 1, (-1/2, 1/2): 1, (1/2, -1/2): 1}, (-1/2, 1/2): {(-3/2, 1/2): 1, (-1/2, -1/2): 1, (-1/2, 3/2): 1, (1/2, 1/2): 1}, (-1/2, 3/2): {}, (1/2, -3/2): {}, (1/2, -1/2): {(-1/2, -1/2): 1, (1/2, -3/2): 1, (1/2, 1/2): 1, (3/2, -1/2): 1}, (1/2, 1/2): {(-1/2, 1/2): 1, (1/2, -1/2): 1, (1/2, 3/2): 1, (3/2, 1/2): 1}, (1/2, 3/2): {}, (3/2, -1/2): {}, (3/2, 1/2): {}} sage: Sandpile(aztec(2),'sink').group_order() 4542720NOTES:

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.

—

**sandlib(selector=None)**

Returns the sandpile identified by

selector. If no argument is given, a description of the sandpiles in the sandlib is printed.INPUT:

selector- identifier or NoneOUTPUT:

sandpile or description

EXAMPLES:

sage: sandlib() Sandpiles in the sandlib: kite : generic undirected graphs with 5 vertices generic : generic digraph with 6 vertices ci1 : complete intersection, non-DAG but equivalent to a DAG 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 gor : Gorenstein but not a complete intersection sage: S = sandlib('gor') sage: S.resolution() 'R <-- R^5 <-- R^5 <-- R^1'

—

**grid(m, n)**

The mxn grid graph. Each nonsink vertex has degree 4.

INPUT:

m,n- positive integersOUTPUT: dictionary for a sandpile with sink named

sink.EXAMPLE:

sage: grid(3,4) {'sink': {}, (1, 1): {'sink': 2, (1, 2): 1, (2, 1): 1}, (1, 2): {'sink': 1, (1, 1): 1, (1, 3): 1, (2, 2): 1}, (1, 3): {'sink': 1, (1, 2): 1, (1, 4): 1, (2, 3): 1}, (1, 4): {'sink': 2, (1, 3): 1, (2, 4): 1}, (2, 1): {'sink': 1, (1, 1): 1, (2, 2): 1, (3, 1): 1}, (2, 2): {(1, 2): 1, (2, 1): 1, (2, 3): 1, (3, 2): 1}, (2, 3): {(1, 3): 1, (2, 2): 1, (2, 4): 1, (3, 3): 1}, (2, 4): {'sink': 1, (1, 4): 1, (2, 3): 1, (3, 4): 1}, (3, 1): {'sink': 2, (2, 1): 1, (3, 2): 1}, (3, 2): {'sink': 1, (2, 2): 1, (3, 1): 1, (3, 3): 1}, (3, 3): {'sink': 1, (2, 3): 1, (3, 2): 1, (3, 4): 1}, (3, 4): {'sink': 2, (2, 4): 1, (3, 3): 1}} sage: S = Sandpile(grid(3,4),'sink') sage: S.group_order() 4140081

—

**random_graph(num_verts, p=1/2, directed=True, weight_max=1)**

A random weighted digraph with a directed spanning tree rooted at . If

directed = False, the only difference is that if is an edge with tail , head , and weight , then appears also. The result is returned as a Sage digraph.INPUT:

num_verts- number of verticesp- probability edges occurdirected- True if directedweight_max- integer maximum for random weightsOUTPUT:

random graph

EXAMPLES:

sage: g = random_graph(6,0.2,True,3) sage: S = Sandpile(g,0) sage: S.show(edge_labels = True)

—

**random_DAG(num_verts, p=1/2, weight_max=1)**

Returns a random directed acyclic graph with

num_vertsvertices. The method starts with the sink vertex and adds vertices one at a time. Each vertex is connected only to only previously defined vertices, and the probability of each possible connection is given by the argumentp. The weight of an edge is a random integer between1andweight_max.INPUT:

num_verts- positive integerp- number between andweight_max– integer greater thanOUTPUT:

directed acyclic graph with sink

EXAMPLES:

sage: S = random_DAG(5, 0.3)

—

**glue_graphs(g, h, glue_g, glue_h)**

Glue two graphs together.

INPUT:

g,h- dictionaries for directed multigraphsglue_h,glue_g- dictionaries for a vertexOUTPUT:

dictionary for a directed multigraph

EXAMPLES:

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) sage: z {0: {}, 'x0': {0: 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': {0: 2}, 'y2': {'y1': 2}, 'y3': {0: 1, 'y2': 1}} sage: S = Sandpile(z,0) sage: S.first_diffs_hilb() [1, 6, 17, 31, 41, 41, 31, 17, 6, 1] sage: S.resolution() 'R <-- R^7 <-- R^21 <-- R^35 <-- R^35 <-- R^21 <-- R^7 <-- R^1'NOTES:

This method makes a dictionary for a graph by combining those for

gandh. The sink ofgis replaced by a vertex that is connected to the vertices ofgas specified byglue_gthe vertices ofhas specified inglue_h. The sink of the glued graph is .Both

glue_gandglue_hare dictionaries with entries of the formv:wwherevis the vertex to be connected to andwis the weight of the connecting edge.

Documentation for each method is available through the Sage online help system:

```
sage: Sandpile.fire_vertex?
Namespace: Interactive
File: /home/davidp/.sage/temp/xyzzy/7883/_home_davidp_math_sandpile_sage_sage_sandpile1_4_sandpile_sage_47.py
Definition: Sandpile.fire_vertex(self, v, config=None)
Docstring:
Fire (topple) a given vertex of a configuration.
INPUT:
- ``v`` - vertex name
- ``config`` - dict
OUTPUT:
dict (configuration)
EXAMPLES::
sage: G = sandlib('generic')
sage: c = G.add(G.max_stable(),G.identity())
sage: G.unstable(c)
[1, 2, 3, 4]
sage: c
{1: 4, 2: 4, 3: 2, 4: 2, 5: 1}
sage: G.fire_vertex(1,c)
NOTES:
This method fires vertex ``v`` in ``config`` provided ``v`` is a
nonsink, unstable vertex. Returns the result (and modifies
``self.config``). The vertex ``v`` is fired only once, even if the
result leaves ``v`` unstable.
```

Note

An alternative to `Sandpile.fire_vertex?` in the preceding code example
would be `S.fire_vertex?`, if `S` is any sandpile.

General Sage documentation can be found at http://sagemath.org/doc/.

Please contact davidp@reed.edu with questions, bug reports, and suggestions for additional features and other improvements.

[BN] | Matthew Baker, Serguei Norine, Riemann-Roch and Abel-Jacobi Theory on a Finite Graph, Advances in Mathematics 215 (2007), 766–788. |

[BTW] | Per Bak, Chao Tang and Kurt Wiesenfeld (1987). Self-organized criticality: an explanation of 1/ƒ noise, Physical Review Letters 60: 381–384 Wikipedia article. |

[H] | Holroyd, Levine, Meszaros, Peres, Propp, Wilson, Chip-Firing and Rotor-Routing on Directed Graphs. The final version of this paper appears in In and out of Equilibrium II, Eds. V. Sidoravicius, M. E. Vares, in the Series Progress in Probability, Birkhauser (2008). |

[PP] | David Perkinson and Jacob Perlman, The algebraic geometry of sandpile groups, preprint (2009). |