*Author: David Perkinson, Reed College*

These notes provide an introduction to Dhar’s abelian sandpile model (ASM) and
to Sage Sandpiles, a collection of tools in Sage for doing sandpile
calculations. For a more thorough introduction to the theory of the ASM, the
papers *Chip-Firing and Rotor-Routing on Directed Graphs* [H], by Holroyd et
al. and *Riemann-Roch and Abel-Jacobi Theory on a Finite Graph* by Baker and
Norine [BN] are recommended.

To describe the ASM, we start with a *sandpile graph*: a directed multigraph
\(\Gamma\) with a vertex \(s\) that is accessible from every vertex (except
possibly \(s\), itself). By *multigraph*, we mean that each edge of \(\Gamma\) is
assigned a nonnegative integer weight. To say \(s\) is *accessible* from some
vertex \(v\) means that there is a sequence of directed edges starting at \(v\) and
ending at \(s\). We call \(s\) the *sink* of the sandpile graph, even though it might have outgoing edges, for reasons that will be made clear in a moment.

We denoted the vertices of \(\Gamma\) by \(V\) and define \(\tilde{V} = V\setminus\{s\}\).

A *configuration* on \(\Gamma\) is an element of \(\mathbb{N}\tilde{V}\), 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 \(\Gamma\) is an element of \(\mathbb{Z}V\), 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 \(c\) is *stable* at a vertex \(v\in\tilde{V}\) if
\(c(v)<\mbox{out-degree}(v)\), and \(c\) itself is stable if it is stable at each
nonsink vertex. Otherwise, \(c\) is *unstable*. If \(c\) is unstable at \(v\), the vertex \(v\) can be *fired*
(*toppled*) by removing \(\mbox{out-degree}(v)\) grains of sand from \(v\) and
adding grains of sand to the neighbors of sand, determined by the weights of
the edges leaving \(v\).

Despite our best intentions, we sometimes consider firing a stable vertex,
resulting in a configuration with a “negative amount” of sand at that vertex.
We may also *reverse-firing* a vertex, absorbing sand from the vertex’s
neighbors.

**Example.** Consider the graph:

All edges have weight \(1\) except for the edge from vertex 1 to vertex 3, which has weight \(2\). If we let \(c=(5,0,1)\) 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 \(c'=(1,1,3)\). Here, \(4\) 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 follows.

```
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') # create the sandpile
sage: S.show(edge_labels=true) # display the graph
Create the configuration:
sage: c = SandpileConfig(S, {1:5, 2:0, 3:1})
sage: S.out_degree()
{1: 4, 2: 2, 3: 2, 'sink': 0}
Fire vertex one:
sage: c.fire_vertex(1)
{1: 1, 2: 1, 3: 3}
The configuration is unchanged:
sage: c
{1: 5, 2: 0, 3: 1}
Repeatedly fire vertices until the configuration becomes stable:
sage: c.stabilize()
{1: 2, 2: 1, 3: 1}
Alternatives:
sage: ~c # shorthand for c.stabilize()
{1: 2, 2: 1, 3: 1}
sage: c.stabilize(with_firing_vector=true)
[{1: 2, 2: 1, 3: 1}, {1: 2, 2: 2, 3: 3}]
```

Since vertex 3 has become unstable after firing vertex 1, it can be fired, which causes vertex 2 to become unstable, etc. 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, \((2,1,1)\). The second component records how many times each vertex fired in the stabilization.

Since the sink is accessible from each nonsink vertex and never fires, 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 \(\Gamma\). The *Laplacian* of \(\Gamma\) is

\[L := D-A\]

where \(D\) is the diagonal matrix of out-degrees of the vertices and \(A\) is the
adjacency matrix whose \((i,j)\)-th entry is the weight of the edge from vertex
\(i\) to vertex \(j\), which we take to be \(0\) if there is no edge. The *reduced
Laplacian*, \(\tilde{L}\), 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 corresponding row of the reduced
Laplacian.

**Example.** (Continued.)

```
sage: S.vertices() # 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 = SandpileConfig(S, [5,0,1])
sage: c
{1: 5, 2: 0, 3: 1}
Firing vertex 1 is the same as subtracting the
corresponding row from the reduced Laplacian:
sage: c.fire_vertex(1).values()
[1, 1, 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 \(c\) is *recurrent* if (i) it is stable, and (ii)
given any configuration \(a\), there is a configuration \(b\) such that
\(c=\mbox{stab}(a+b)\), the stabilization of \(a+b\).

The *maximal-stable* configuration, denoted \(c_{\mathrm{max}}\) is defined by
\(c_{\mathrm{max}}(v)=\mbox{out-degree}(v)-1\) for all nonsink vertices \(v\). It is clear that \(c_{\mathrm{max}}\) is recurrent. Further, it is not hard to see that a configuration is recurrent if and only if it has the form \(\mbox{stab}(a+c_{\mathrm{max}})\) for some configuration \(a\).

**Example.** (Continued.)

```
sage: S.recurrents(verbose=false)
[[3, 1, 1], [2, 1, 1], [3, 1, 0]]
sage: c = SandpileConfig(S, [2,1,1])
sage: c
{1: 2, 2: 1, 3: 1}
sage: c.is_recurrent()
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 = SandpileConfig(S, [1,0,0])
sage: x + S.max_stable()
{1: 4, 2: 1, 3: 1}
Use & to add and stabilize:
sage: c = x & S.max_stable()
sage: c
{1: 3, 2: 1, 3: 0}
sage: c.is_recurrent()
True
Note the various ways of performing addition + stabilization:
sage: m = S.max_stable()
sage: (x + m).stabilize() == ~(x + m)
True
sage: (x + m).stabilize() == x & m
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 \(b\) is the burning configuration, \(\sigma\) is its
script, and \(\tilde{L}\) is the reduced Laplacian, then \(\sigma\,\tilde{L} = b\).
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 \(\sigma\).

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 \(\Gamma\), denoted
\(\mathcal{S}(\Gamma)\). Its identity element is usually not the all-zero
configuration (again, only in the case that \(\Gamma\) is a DAG). So finding the
identity element is an interesting problem.

Let \(n=|V|-1\) and fix an ordering of the nonsink vertices. Let \(\mathcal{\tilde{L}}\subset\mathbb{Z}^n\) denote the column-span of \(\tilde{L}^t\), the transpose of the reduced Laplacian. It is a theorem that

\[\mathcal{S}(\Gamma)\approx \mathbb{Z}^n/\mathcal{\tilde{L}}.\]

Thus, the number of elements of the sandpile group is \(\det{\tilde{L}}\), 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.invariant_factors()
[1, 1, 3]
sage: S.reduced_laplacian().dense_matrix().smith_form()
(
[1 0 0] [ 0 0 1] [3 1 4]
[0 1 0] [ 1 0 0] [4 1 6]
[0 0 3], [ 0 1 -1], [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: i & m == 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 precise definition, but can be
loosely taken to describe a system that naturally evolves to a state that is
barely stable and such that the instabilities are described by a power law.
In practice, *self-organized criticality* is often 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 \(2\). Thus, every nonsink
vertex has out-degree \(4\).

Imagine repeatedly dropping grains of sand on and empty grid graph, allowing the sandpile to stabilize in between. At first there is little activity, but as time goes on, the size and extent of the avalanche caused by a single grain of sand becomes hard to predict. Computer experiments—I do not think there is a proof, yet—indicate that the distribution of avalanche sizes obeys a power law with exponent -1. In the example below, the size of an avalanche is taken to be the sum of the number of times each vertex fires.

**Example (distribution of avalanche sizes).**

```
sage: S = grid_sandpile(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(10000): # long time (15s on sage.math, 2012)
....: 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)]) # long time
sage: p.axes_labels(['log(N)','log(D(N))']) # long time
sage: p # long time
Graphics object consisting of 1 graphics primitive
```

Note: In the above code, `m.stabilize(true)` returns a list consisting of the
stabilized configuration and the firing vector. (Omitting `true` would give
just the stabilized configuration.)

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

A *divisor* on \(\Gamma\) is an element of the free abelian group on its
vertices, including the sink. Suppose, as above, that the \(n+1\) vertices of
\(\Gamma\) have been ordered, and that \(\mathcal{L}\) is the column span of the
transpose of the Laplacian. A divisor is then identified with an element
\(D\in\mathbb{Z}^{n+1}\) and two divisors are *linearly equivalent* if they
differ by an element of \(\mathcal{L}\). A divisor \(E\) is *effective*, written
\(E\geq0\), if \(E(v)\geq0\) for each \(v\in V\), i.e., if \(E\in\mathbb{N}^{n+1}\).
The *degree* of a divisor, \(D\), is \(deg(D) := \sum_{v\in V}D(v)\). 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 \(D\), denoted \(|D|\), is the
collection of effective divisors linearly equivalent to \(D.\)

To describe the Riemann-Roch theorem in this context, suppose that \(\Gamma\) is
an undirected, unweighted graph. The *dimension*, \(r(D)\) of the linear system
\(|D|\) is \(-1\) if \(|D|=\emptyset\) and otherwise is the greatest integer \(s\) such
that \(|D-E|\neq0\) for all effective divisors \(E\) of degree \(s\). Define the
*canonical divisor* by \(K=\sum_{v\in V}(\deg(v)-2)v\) and the *genus* by \(g =
\#(E) - \#(V) + 1\). The Riemann-Roch theorem says that for any divisor \(D\),

\[r(D)-r(K-D)=\deg(D)+1-g.\]

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

```
sage: G = complete_sandpile(5) # the sandpile on the complete graph with 5 vertices
The genus (num_edges method counts each undirected edge twice):
sage: g = G.num_edges()/2 - G.num_verts() + 1
A divisor on the graph:
sage: D = SandpileDivisor(G, [1,2,2,0,2])
Verify the Riemann-Roch theorem:
sage: K = G.canonical_divisor()
sage: D.r_of_D() - (K - D).r_of_D() == D.deg() + 1 - g # optional - 4ti2
True
The effective divisors linearly equivalent to D:
sage: [E.values() for E in D.effective_div()] # optional - 4ti2
[[0, 1, 1, 4, 1], [4, 0, 0, 3, 0], [1, 2, 2, 0, 2]]
The nonspecial divisors up to linear equivalence (divisors of degree
g-1 with empty linear systems)
sage: N = G.nonspecial_divisors()
sage: [E.values() for E in N[:5]] # the first few
[[-1, 2, 1, 3, 0],
[-1, 0, 3, 1, 2],
[-1, 2, 0, 3, 1],
[-1, 3, 1, 2, 0],
[-1, 2, 0, 1, 3]]
sage: len(N)
24
sage: len(N) == G.h_vector()[-1]
True
```

Fix a divisor \(D\). There are at least two natural graphs associated with linear system associated with \(D\). First, consider the directed graph with vertex set \(|D|\) and with an edge from vertex \(E\) to vertex \(F\) if \(F\) is attained from \(E\) by firing a single unstable vertex.

```
sage: S = Sandpile(graphs.CycleGraph(6),0)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: D.is_alive()
True
sage: eff = D.effective_div() # optional - 4ti2
sage: firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01,iterations=500) # optional - 4ti2
```

The second graph has the same set of vertices but with an edge from \(E\) to \(F\) if \(F\) is obtained from \(E\) by firing all unstable vertices of \(E\).

```
sage: S = Sandpile(graphs.CycleGraph(6),0)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div() # optional - 4ti2
sage: parallel_firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01,iterations=500) # optional - 4ti2
```

Note that in each of the examples, above, starting at any divisor in the linear
system and following edges, one is eventually led into a cycle of length 6
(cycling the divisor (1,1,1,1,2,0)). Thus, `D.alive()` returns `True`. In
Sage, one would be able to rotate the above figures to get a better idea of the
structure.

Let \(n=|V|-1\), and fix an ordering on the nonsink vertices of \(\Gamma\). let
\(\tilde{\mathcal{L}}\subset\mathbb{Z}^n\) denote the column-span of
\(\tilde{L}^t\), the transpose of the reduced Laplacian. Label vertex \(i\) with the
indeterminate \(x_i\), and let \(\mathbb{C}[\Gamma_s] = \mathbb{C}[x_1,\dots,x_n]\).
(Here, \(s\) denotes the sink vertex of \(\Gamma\).) The *sandpile ideal* or
*toppling ideal*, first studied by Cori, Rossin, and Salvy [CRS] for undirected graphs, is the lattice ideal for \(\tilde{\mathcal{L}}\):

\[I = I(\Gamma_s) := \{x^u-x^v: u-v\in
\tilde{\mathcal{L}}\}\subset\mathbb{C}[\Gamma_s],\]

where \(x^u := \prod_{i=1}^nx^{u_i}\) for \(u\in\mathbb{Z}^n\).

For each \(c\in\mathbb{Z}^n\) define \(t(c) = x^{c^+} - x^{c^-}\) where \(c^+_i=\max\{c_i,0\}\) and \(c^-=\max\{-c_i,0\}\) so that \(c=c^+-c^-\). Then, for each \(\sigma\in\mathbb{Z}^n\), define \(T(\sigma) = t(\tilde{L}^t\sigma)\). It then turns out that

\[I = (T(e_1),\dots,T(e_n),x^b-1)\]

where \(e_i\) is the \(i\)-th standard basis vector and \(b\) is any burning configuration.

The affine coordinate ring, \(\mathbb{C}[\Gamma_s]/I,\) is isomorphic to the group algebra of the sandpile group, \(\mathbb{C}[\mathcal{S}(\Gamma)].\)

The standard term-ordering on \(\mathbb{C}[\Gamma_s]\) is graded reverse lexigraphical order with \(x_i>x_j\) if vertex \(v_i\) is further from the sink than vertex \(v_j\). (There are choices to be made for vertices equidistant from the sink). If \(\sigma_b\) is the script for a burning configuration (not necessarily minimal), then

\[\{T(\sigma): \sigma\leq\sigma_b\}\]

is a Groebner basis for \(I\).

Now let \(\mathbb{C}[\Gamma]=\mathbb{C}[x_0,x_1,\dots,x_n]\), where \(x_0\)
corresponds to the sink vertex. The *homogeneous sandpile ideal*, denoted
\(I^h\), is obtaining by homogenizing \(I\) with respect to \(x_0\). Let \(L\) be the
(full) Laplacian, and \(\mathcal{L}\subset\mathbb{Z}^{n+1}\) be the column span of
its transpose, \(L^t.\) Then \(I^h\) is the lattice ideal for \(\mathcal{L}\):

\[I^h = I^h(\Gamma) := \{x^u-x^v: u-v \in\mathcal{L}\}\subset\mathbb{C}[\Gamma].\]

This ideal can be calculated by saturating the ideal

\[(T(e_i): i=0,\dots n)\]

with respect to the product of the indeterminates: \(\prod_{i=0}^nx_i\) (extending the \(T\) operator in the obvious way). A Groebner basis with respect to the degree lexicographic order describe above (with \(x_0\) the smallest vertex), is obtained by homogenizing each element of the Groebner basis for the non-homogeneous sandpile ideal with respect to \(x_0.\)

**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()
Multivariate Polynomial Ring in x5, x4, x3, x2, x1, x0 over Rational Field
The homogeneous sandpile ideal:
sage: S.ideal()
Ideal (x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0,
x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2) of Multivariate Polynomial Ring
in x5, x4, x3, x2, x1, x0 over Rational Field
The generators of the ideal:
sage: S.ideal(true)
[x2 - x0,
x3^2 - x5*x0,
x5*x3 - x0^2,
x4^2 - x3*x1,
x5^2 - x3*x0,
x1^3 - x4*x3*x0,
x4*x1^2 - x5*x0^2]
Its resolution:
sage: S.resolution() # long time
'R^1 <-- R^7 <-- R^19 <-- R^25 <-- R^16 <-- R^4'
and Betti table:
sage: S.betti() # long time
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.h_vector()
[1, 4, 6, 4]
sage: x = [i.deg() for i in S.superstables()]
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 \(I\) is

\[Z(I) = \{p\in\mathbb{C}^n: f(p)=0\mbox{ for all $f\in I$}\},\]

the set of simultaneous zeros of the polynomials in \(I.\) Letting \(S^1\) denote the unit circle in the complex plane, \(Z(I)\) is a finite subgroup of \(S^1\times\dots\times S^1\subset\mathbb{C}^n\), isomorphic to the sandpile group. The zero set is actually linearly isomorphic to a faithful representation of the sandpile group on \(\mathbb{C}^n.\)

**Example.** (Continued.)

```
sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0)
sage: S.ideal().gens()
[x1^2 - x2^2, x1*x2^3 - x0^4, x2^5 - x1*x0^4]
Approximation to the zero set (setting ``x_0 = 1``):
sage: S.solve()
[[-0.707107 + 0.707107*I, 0.707107 - 0.707107*I],
[-0.707107 - 0.707107*I, 0.707107 + 0.707107*I],
[-I, -I],
[I, I],
[0.707107 + 0.707107*I, -0.707107 - 0.707107*I],
[0.707107 - 0.707107*I, -0.707107 + 0.707107*I],
[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, \(I^h\), has a free resolution graded by the
divisors on \(\Gamma\) modulo linear equivalence. (See the section on
*Discrete Riemann Surfaces* for the language of
divisors and linear equivalence.) Let
\(S=\mathbb{C}[\Gamma]=\mathbb{C}[x_0,\dots,x_n]\), as above, and let
\(\mathfrak{S}\) denote the group of divisors modulo rational equivalence. Then
\(S\) is graded by \(\mathfrak{S}\) by letting \(\deg(x^c)= c\in\mathfrak{S}\) for
each monomial \(x^c\). The minimal free resolution of \(I^h\) has the form

\[ 0\leftarrow I^h
\leftarrow\oplus_{D\in\mathfrak{S}}S(-D)^{\beta_{0,D}}\leftarrow\oplus_{D\in\mathfrak{S}}S(-D)^{\beta_{1,D}}
\leftarrow\dots\leftarrow\oplus_{D\in\mathfrak{S}}S(-D)^{\beta_{r,D}}\leftarrow0.\]

where the \(\beta_{i,D}\) are the *Betti numbers* for \(I^h\).

For each divisor class \(D\in\mathfrak{S}\), define a simplicial complex,

\[\Delta_D := \{I\subseteq\{0,\dots,n\}: I\subseteq\mbox{supp}(E)\mbox{ for some}\
E\in |D|\}.\]

The Betti number \(\beta_{i,D}\) equals the dimension over \(\mathbb{C}\) of the \(i\)-th reduced homology group of \(\Delta_D\):

\[\beta_{i,D} = \dim_{\mathbb{C}}\tilde{H}_i(\Delta_D;\mathbb{C}).\]

```
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() # optional - 4ti2
sage: p[0] # optional - 4ti2
[{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] # optional - 4ti2
sage: D.effective_div() # optional - 4ti2
[{0: 0, 1: 1, 2: 1, 3: 0}, {0: 0, 1: 0, 2: 0, 3: 2}]
sage: [E.support() for E in D.effective_div()] # optional - 4ti2
[[1, 2], [3]]
sage: D.Dcomplex() # optional - 4ti2
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 2), (3,)}
sage: D.Dcomplex().homology() # optional - 4ti2
{0: Z, 1: 0}
The minimal free resolution:
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) # optional - 4ti2
11
The degrees and ranks of the homology groups for each element of the list p
(compare with the Betti table, above):
sage: [[sum(d[0].values()),d[1].betti()] for d in p] # optional - 4ti2
[[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[5, {0: 1, 1: 0, 2: 1}]]
```

NOTE: in the previous section note that the resolution always has length \(n\) since the ideal is Cohen-Macaulay.

To do.

To do.

Warning

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

To install 4ti2:

- Go to the Sage website and
- look for the
*precise name*of the 4ti2 package and install it according to the instructions given there. For instance, suppose the package is named 4ti2.p0.spkg. Install the package with the following command from a UNIX shell prompt:

```
sage -i 4ti2.p0
```

There are three main classes for sandpile structures in Sage: `Sandpile`,
`SandpileConfig`, and `SandpileDivisor`. Initialization for `Sandpile`
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 \(v\) is a dictionary
consisting of pairs: `vertex: weight`. Each pair represents a directed edge
emanating from \(v\) 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.CycleGraph(5) sage: S = Sandpile(g, 0) sage: type(g) <class 'sage.graphs.graph.Graph'>

To see the types of built-in graphs, type `graphs.`, including the period,
and hit TAB.

a Sage digraph:

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

See sage.graphs.graph_generators 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 are summaries of `Sandpile`, `SandpileConfig`, and `SandpileDivisor` methods
(functions). Each summary is followed by a list of complete descriptions of
the methods. There are many more methods available for a Sandpile, e.g.,
those inherited from the class DiGraph. To see them all, enter
`dir(Sandpile)` or type `Sandpile.`, including the period, and hit TAB.

**Summary of methods.**

*all_k_config(k)*— The configuration with all values set to k.*all_k_div(k)*— The divisor with all values set to k.*betti(verbose=True)*— The Betti table for the homogeneous sandpile ideal.*betti_complexes()*— The divisors with nonempty linear systems along with their simplicial complexes.*burning_config()*— A minimal burning configuration.*burning_script()*— A script for the minimal burning configuration.*canonical_divisor()*— The canonical divisor (for undirected graphs).*dict()*— A dictionary of dictionaries representing a directed graph.*groebner()*— Groebner basis for the homogeneous sandpile ideal with respect to the standard sandpile ordering.*group_order()*— The size of the sandpile group.*h_vector()*— The first differences of the Hilbert function of the homogeneous sandpile ideal.*hilbert_function()*— The Hilbert function of the homogeneous sandpile ideal.*ideal(gens=False)*— The saturated, homogeneous sandpile ideal.*identity()*— The identity configuration.*in_degree(v=None)*— The in-degree of a vertex or a list of all in-degrees.*invariant_factors()*— The invariant factors of the sandpile group (a finite abelian group).*is_undirected()*—`True`if`(u,v)`is and edge if and only if`(v,u)`is an edges, each edge with the same weight.*laplacian()*— The Laplacian matrix of the graph.*max_stable()*— The maximal stable configuration.*max_stable_div()*— The maximal stable divisor.*max_superstables(verbose=True)*— The maximal superstable configurations.*min_recurrents(verbose=True)*— The minimal recurrent elements.*nonsink_vertices()*— The names of the nonsink vertices.*nonspecial_divisors(verbose=True)*— The nonspecial divisors (only for undirected graphs).*num_edges()*— The number of edges.*num_verts()*— The number of vertices.*out_degree(v=None)*— 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.*recurrents(verbose=True)*— The list of recurrent configurations.*reduced_laplacian()*— The reduced Laplacian matrix of the graph.*reorder_vertices()*— Create a copy of the sandpile but with the vertices reordered.*resolution(verbose=False)*— The minimal free resolution of the homogeneous sandpile ideal.*ring()*— The ring containing the homogeneous sandpile ideal.*show(kwds)*— Draws the graph.*show3d(kwds)*— Draws the graph.*sink()*— The identifier for the sink vertex.*solve()*— Approximations of the complex affine zeros of the sandpile ideal.*superstables(verbose=True)*— The list of superstable configurations.*symmetric_recurrents(orbits)*— The list of symmetric recurrent configurations.*unsaturated_ideal()*— The unsaturated, homogeneous sandpile ideal.*version()*— The version number of Sage Sandpiles.*vertices(key=None, boundary_first=False)*— List of the vertices.*zero_config()*— The all-zero configuration.*zero_div()*— The all-zero divisor.

**Complete descriptions of Sandpile methods.**

—

**all_k_config(k)**

The configuration with all values set to k.

INPUT:

k- integerOUTPUT:

SandpileConfig

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 with all values set to k.

INPUT:

k- integerOUTPUT:

SandpileDivisor

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

EXAMPLES:

sage: S = sandlib('generic') sage: S.betti() # long time 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 sage: S.betti(False) # long time [1, 7, 19, 25, 16, 4]

—

**betti_complexes()**

A list of all the divisors with nonempty linear systems whose corresponding simplicial complexes have nonzero homology in some dimension. Each such divisor 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() # optional - 4ti2 sage: p[0] # optional - 4ti2 [{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^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) # optional - 4ti2 11 sage: p[0][1].homology() # optional - 4ti2 {0: Z, 1: 0} sage: p[-1][1].homology() # optional - 4ti2 {0: 0, 1: 0, 2: Z}

—

**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: 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]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 b is the burning configuration, sigma is its script, and tilde{L} is the reduced Laplacian, then sigma * tilde{L} = b. 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 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()**

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: 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]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 b is the burning configuration, s is its script, and L_{mathrm{red}} is the reduced Laplacian, then s * L_{mathrm{red}}= b. 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 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()**

The canonical divisor: the divisor

deg(v)-2grains of sand on each vertex. Only for undirected graphs.INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = complete_sandpile(4) sage: S.canonical_divisor() {0: 1, 1: 1, 2: 1, 3: 1}

—

**dict()**

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

—

**groebner()**

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() [x4*x1^2 - x5*x0^2, x1^3 - x4*x3*x0, x5^2 - x3*x0, x4^2 - x3*x1, x5*x3 - x0^2, x3^2 - x5*x0, x2 - x0]

—

**group_order()**

The size of the sandpile group.

INPUT:

None

OUTPUT:

int

EXAMPLES:

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

—

**h_vector()**

The first differences of the Hilbert function of the homogeneous sandpile ideal. It lists the number of superstable configurations in each degree.

INPUT:

None

OUTPUT:

list of nonnegative integers

EXAMPLES:

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

—

**hilbert_function()**

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(gens=False)**

The saturated, homogeneous sandpile ideal (or its generators if

gens=True).INPUT:

verbose(optional) - booleanOUTPUT:

ideal or, optionally, the generators of an ideal

EXAMPLES:

sage: S = sandlib('generic') sage: S.ideal() Ideal (x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0, x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2) of Multivariate Polynomial Ring in x5, x4, x3, x2, x1, x0 over Rational Field sage: S.ideal(True) [x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0, x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2] sage: S.ideal().gens() # another way to get the generators [x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0, x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2]

—

**identity()**

The identity configuration.

INPUT:

None

OUTPUT:

dict (the identity configuration)

EXAMPLES:

sage: S = sandlib('generic') sage: e = S.identity() sage: x = e & S.max_stable() # stable addition sage: x {1: 2, 2: 2, 3: 1, 4: 1, 5: 1} sage: x == S.max_stable() True

—

**in_degree(v=None)**

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

INPUT:

v- vertex name or NoneOUTPUT:

integer or dict

EXAMPLES:

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

—

**invariant_factors()**

The invariant factors of the sandpile group (a finite abelian group).

INPUT:

None

OUTPUT:

list of integers

EXAMPLES:

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

—

**is_undirected()**

Trueif(u,v)is and edge if and only if(v,u)is an edges, each edge with the same weight.INPUT:

None

OUTPUT:

boolean

EXAMPLES:

sage: complete_sandpile(4).is_undirected() True sage: sandlib('gor').is_undirected() False

—

**laplacian()**

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]NOTES:

The function

laplacian_matrixshould be avoided. It returns the indegree version of the laplacian.

—

**max_stable()**

The maximal stable configuration.

INPUT:

None

OUTPUT:

SandpileConfig (the maximal stable configuration)

EXAMPLES:

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

—

**max_stable_div()**

The maximal stable divisor.

INPUT:

SandpileDivisor

OUTPUT:

SandpileDivisor (the maximal stable divisor)

EXAMPLES:

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

—

**max_superstables(verbose=True)**

The maximal superstable configurations. If the underlying graph is undirected, these are the superstables of highest degree. If

verboseisFalse, the configurations are converted to lists of integers.INPUT:

verbose(optional) - booleanOUTPUT:

list (of maximal superstables)

EXAMPLES:

sage: S=sandlib('riemann-roch2') sage: S.max_superstables() [{1: 1, 2: 1, 3: 1}, {1: 0, 2: 0, 3: 2}] sage: S.superstables(False) [[0, 0, 0], [1, 0, 1], [1, 0, 0], [0, 1, 1], [0, 1, 0], [1, 1, 0], [0, 0, 1], [1, 1, 1], [0, 0, 2]] sage: S.h_vector() [1, 3, 4, 1]

—

**min_recurrents(verbose=True)**

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(optional) - booleanOUTPUT:

list of SandpileConfig

EXAMPLES:

sage: S=sandlib('riemann-roch2') sage: S.min_recurrents() [{1: 0, 2: 0, 3: 1}, {1: 1, 2: 1, 3: 0}] sage: S.min_recurrents(False) [[0, 0, 1], [1, 1, 0]] sage: S.recurrents(False) [[1, 1, 2], [0, 1, 1], [0, 1, 2], [1, 0, 1], [1, 0, 2], [0, 0, 2], [1, 1, 1], [0, 0, 1], [1, 1, 0]] sage: [i.deg() for i in S.recurrents()] [4, 2, 3, 2, 3, 2, 3, 1, 2]

—

**nonsink_vertices()**

The names of the nonsink vertices.

INPUT:

None

OUTPUT:

None

EXAMPLES:

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

—

**nonspecial_divisors(verbose=True)**

The nonspecial divisors: those divisors of degree

g-1with empty linear system. The term is only defined for undirected graphs. Here,g = |E| - |V| + 1is the genus of the graph. IfverboseisFalse, the divisors are converted to lists of integers.INPUT:

verbose(optional) - booleanOUTPUT:

list (of divisors)

EXAMPLES:

sage: S = complete_sandpile(4) sage: ns = S.nonspecial_divisors() # optional - 4ti2 sage: D = ns[0] # optional - 4ti2 sage: D.values() # optional - 4ti2 [-1, 1, 0, 2] sage: D.deg() # optional - 4ti2 2 sage: [i.effective_div() for i in ns] # optional - 4ti2 [[], [], [], [], [], []]

—

**num_edges()**

The number of edges.

EXAMPLES:

sage: G = graphs.PetersenGraph() sage: G.size() 15

—

**num_verts()**

The number of vertices. Note that len(G) returns the number of vertices in G also.

EXAMPLES:

sage: G = graphs.PetersenGraph() sage: G.order() 10 sage: G = graphs.TetrahedralGraph() sage: len(G) 4

—

**out_degree(v=None)**

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

INPUT:

v(optional) - vertex nameOUTPUT:

integer or dict

EXAMPLES:

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

—

**points()**

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()**

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

—

**recurrents(verbose=True)**

The list of recurrent configurations. If

verboseisFalse, the configurations are converted to 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(verbose=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()**

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.

—

**reorder_vertices()**

Create a copy of the sandpile but with the vertices ordered according to their distance from the sink, from greatest to least.

INPUT:

None

OUTPUT:

Sandpile

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: T = S.reorder_vertices() sage: T.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, 4: 1}, 4: {}}

—

**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 = sandlib('gor') sage: S.resolution() '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]

—

**ring()**

The ring containing the homogeneous sandpile ideal.

INPUT:

None

OUTPUT:

ring

EXAMPLES:

sage: S = sandlib('generic') sage: S.ring() Multivariate Polynomial Ring in x5, x4, x3, x2, x1, x0 over Rational Field sage: S.ring().gens() (x5, x4, x3, x2, x1, x0)NOTES:

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)**

Draws the graph.

INPUT:

kwds- arguments passed to the show method for Graph or DiGraphOUTPUT:

None

EXAMPLES:

sage: S = sandlib('generic') sage: S.show() sage: S.show(graph_border=True, edge_labels=True)

—

**show3d(kwds)**

Draws the graph.

INPUT:

kwds- arguments passed to the show method for Graph or DiGraphOUTPUT:

None

EXAMPLES:

sage: S = sandlib('generic') sage: S.show3d()

—

**sink()**

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 = grid_sandpile(2,2) sage: H.sink() 'sink' sage: type(H.sink()) <type 'str'>

—

**solve()**

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 + 0.707107*I, 0.707107 - 0.707107*I], [-0.707107 - 0.707107*I, 0.707107 + 0.707107*I], [-I, -I], [I, I], [0.707107 + 0.707107*I, -0.707107 - 0.707107*I], [0.707107 - 0.707107*I, -0.707107 + 0.707107*I], [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().

—

**superstables(verbose=True)**

The list of superstable configurations as dictionaries if

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

verbose(optional) - 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]]

—

**symmetric_recurrents(orbits)**

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().gens() [x1^3 - x4*x3*x0, x2^3 - x5*x3*x0, x3^2 - x5*x2, x4^2 - x3*x1, x5^2 - x3*x2] sage: S.ideal().gens() [x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0, x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2]

—

**version()**

The version number of Sage Sandpiles.

INPUT:

None

OUTPUT:

string

EXAMPLES:

sage: S = sandlib('generic') sage: S.version() Sage Sandpiles Version 2.3

—

**vertices(key=None, boundary_first=False)**

A list of the vertices.

INPUT:

key- default:None- a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm.boundary_first- default:False- ifTrue, return the boundary vertices first.OUTPUT:

The vertices of the list.

Warning: There is always an attempt to sort the list before returning the result. However, since any object may be a vertex, there is no guarantee that any two vertices will be comparable. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sort, use the

keyargument, as illustrated in the examples below.EXAMPLES:

sage: P = graphs.PetersenGraph() sage: P.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

—

**zero_config()**

The all-zero configuration.

INPUT:

None

OUTPUT:

SandpileConfig

EXAMPLES:

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

—

**zero_div()**

The all-zero divisor.

INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

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

—

**Summary of methods.**

*+*— Addition of configurations.*&*— The stabilization of the sum.*greater-equal*—`True`if every component of`self`is at least that of`other`.*greater*—`True`if every component of`self`is at least that of`other`and the two configurations are not equal.*~*— The stabilized configuration.*less-equal*—`True`if every component of`self`is at most that of`other`.*less*—`True`if every component of`self`is at most that of`other`and the two configurations are not equal.***— The recurrent element equivalent to the sum.*^*— Exponentiation for the *-operator.*-*— The additive inverse of the configuration.*-*— Subtraction of configurations.*add_random()*— Add one grain of sand to a random nonsink vertex.*deg()*— The degree of the configuration.*dualize()*— The difference between the maximal stable configuration and the configuration.*equivalent_recurrent(with_firing_vector=False)*— The equivalent recurrent configuration.*equivalent_superstable(with_firing_vector=False)*— The equivalent superstable configuration.*fire_script(sigma)*— Fire the script`sigma`, i.e., fire each vertex the indicated number of times.*fire_unstable()*— Fire all unstable vertices.*fire_vertex(v)*— Fire the vertex`v`.*is_recurrent()*—`True`if the configuration is recurrent.*is_stable()*—`True`if stable.*is_superstable()*—`True`if`config`is superstable.*is_symmetric(orbits)*— Is the configuration constant over the vertices in each sublist of`orbits`?*order()*— The order of the recurrent element equivalent to`config`.*sandpile()*— The configuration’s underlying sandpile.*show(sink=True,colors=True,heights=False,directed=None,kwds)*— Show the configuration.*stabilize(with_firing_vector=False)*— The stabilized configuration. Optionally returns the corresponding firing vector.*support()*— Keys of the nonzero values of the dictionary.*unstable()*— List of the unstable vertices.*values()*— The values of the configuration as a list.

**Complete descriptions of SandpileConfig methods.**

—

**+**

Addition of configurations.

INPUT:

other- SandpileConfigOUTPUT:

sum of

selfandotherEXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [3,2]) sage: c + d {1: 4, 2: 4}

—

**&**

The stabilization of the sum.

INPUT:

other- SandpileConfigOUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: c = SandpileConfig(S, [1,0,0]) sage: c + c # ordinary addition {1: 2, 2: 0, 3: 0} sage: c & c # add and stabilize {1: 0, 2: 1, 3: 0} sage: c*c # add and find equivalent recurrent {1: 1, 2: 1, 3: 1} sage: ~(c + c) == c & c True

—

**>=**

Trueif every component ofselfis at least that ofother.INPUT:

other- SandpileConfigOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [2,3]) sage: e = SandpileConfig(S, [2,0]) sage: c >= c True sage: d >= c True sage: c >= d False sage: e >= c False sage: c >= e False

—

**>**

Trueif every component ofselfis at least that ofotherand the two configurations are not equal.INPUT:

other- SandpileConfigOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [1,3]) sage: c > c False sage: d > c True sage: c > d False

—

**~**

The stabilized configuration.

INPUT:

None

OUTPUT:

SandpileConfigEXAMPLES:

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

—

**<=**

Trueif every component ofselfis at most that ofother.INPUT:

other- SandpileConfigOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [2,3]) sage: e = SandpileConfig(S, [2,0]) sage: c <= c True sage: c <= d True sage: d <= c False sage: c <= e False sage: e <= c False

—

**<**

Trueif every component ofselfis at most that ofotherand the two configurations are not equal.INPUT:

other- SandpileConfigOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [2,3]) sage: c < c False sage: c < d True sage: d < c False

—

*****

The recurrent element equivalent to the sum.

INPUT:

other- SandpileConfigOUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: c = SandpileConfig(S, [1,0,0]) sage: c + c # ordinary addition {1: 2, 2: 0, 3: 0} sage: c & c # add and stabilize {1: 0, 2: 1, 3: 0} sage: c*c # add and find equivalent recurrent {1: 1, 2: 1, 3: 1} sage: (c*c).is_recurrent() True sage: c*(-c) == S.identity() True

—

**^**

The recurrent element equivalent to the sum of the configuration with itself

ktimes. Ifkis negative, do the same for the negation of the configuration. Ifkis zero, return the identity of the sandpile group.INPUT:

k- SandpileConfigOUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: c = SandpileConfig(S, [1,0,0]) sage: c^3 {1: 1, 2: 1, 3: 0} sage: (c + c + c) == c^3 False sage: (c + c + c).equivalent_recurrent() == c^3 True sage: c^(-1) {1: 1, 2: 1, 3: 0} sage: c^0 == S.identity() True

—

**_**

The additive inverse of the configuration.

INPUT:

None

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: -c {1: -1, 2: -2}

—

**-**

Subtraction of configurations.

INPUT:

other- SandpileConfigOUTPUT:

sum of

selfandotherEXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: d = SandpileConfig(S, [3,2]) sage: c - d {1: -2, 2: 0}

—

**add_random()**

Add one grain of sand to a random nonsink vertex.

INPUT:

None

OUTPUT:

SandpileConfig

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 = grid_sandpile(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))'])

—

**deg()**

The degree of the configuration.

INPUT:

None

OUTPUT:

integer

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: c.deg() 3

—

**dualize()**

The difference between the maximal stable configuration and the configuration.

INPUT:

None

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) 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

—

**equivalent_recurrent(with_firing_vector=False)**

The recurrent configuration equivalent to the given configuration. Optionally returns the corresponding firing vector.

INPUT:

with_firing_vector(optional) - booleanOUTPUT:

SandpileConfigor[SandpileConfig, firing_vector]EXAMPLES:

sage: S = sandlib('generic') sage: c = SandpileConfig(S, [0,0,0,0,0]) 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() TrueNOTES:

Let L be the reduced laplacian, c the initial configuration, r the returned configuration, and f the firing vector. Then r = c - f * L.

—

**equivalent_superstable(with_firing_vector=False)**

The equivalent superstable configuration. Optionally returns the corresponding firing vector.

INPUT:

with_firing_vector(optional) - booleanOUTPUT:

SandpileConfigor[SandpileConfig, firing_vector]EXAMPLES:

sage: S = sandlib('generic') 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() TrueNOTES:

Let L be the reduced laplacian, c the initial configuration, s the returned configuration, and f the firing vector. Then s = c - f * L.

—

**fire_script(sigma)**

Fire the script

sigma, i.e., fire each vertex the indicated number of times.INPUT:

sigma- SandpileConfig or (list or dict representing a SandpileConfig)OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) 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

—

**fire_unstable()**

Fire all unstable vertices.

INPUT:

None

OUTPUT:

SandpileConfig

EXAMPLES:

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

—

**fire_vertex(v)**

Fire the vertex

v.INPUT:

v- vertexOUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: c = SandpileConfig(S, [1,2]) sage: c.fire_vertex(2) {1: 2, 2: 0}

—

**is_recurrent()**

Trueif the configuration is recurrent.INPUT:

None

OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.identity().is_recurrent() True sage: S.zero_config().is_recurrent() False

—

**is_stable()**

Trueif stable.INPUT:

None

OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.max_stable().is_stable() True sage: (S.max_stable() + S.max_stable()).is_stable() False sage: (S.max_stable() & S.max_stable()).is_stable() True

—

**is_superstable()**

Trueifconfigis superstable, i.e., whether its dual is recurrent.INPUT:

None

OUTPUT:

boolean

EXAMPLES:

sage: S = sandlib('generic') sage: S.zero_config().is_superstable() True

—

**is_symmetric(orbits)**

This function checks if the values of the configuration are constant over the vertices in each sublist of

orbits.INPUT:

orbits- 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 = SandpileConfig(S, [1, 2, 2, 3]) sage: c.is_symmetric([[2,3]]) True

—

**order()**

The order of the recurrent element equivalent to

config.INPUT:

config- configurationOUTPUT:

integer

EXAMPLES:

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

—

**sandpile()**

The configuration’s underlying sandpile.

INPUT:

None

OUTPUT:

Sandpile

EXAMPLES:

sage: S = sandlib('genus2') sage: c = S.identity() sage: c.sandpile() Digraph on 4 vertices sage: c.sandpile() == S True

—

**show(sink=True,colors=True,heights=False,directed=None,kwds)**

Show the configuration.

INPUT:

sink- whether to show the sinkcolors- whether to color-code the amount of sand on each vertexheights- whether to label each vertex with the amount of sandkwds- arguments passed to the show method for Graphdirected- whether to draw directed edgesOUTPUT:

None

EXAMPLES:

sage: S=sandlib('genus2') sage: c=S.identity() sage: S=sandlib('genus2') sage: c=S.identity() sage: c.show() sage: c.show(directed=False) sage: c.show(sink=False,colors=False,heights=True)

—

**stabilize(with_firing_vector=False)**

The stabilized configuration. Optionally returns the corresponding firing vector.

INPUT:

with_firing_vector(optional) - booleanOUTPUT:

SandpileConfigor[SandpileConfig, firing_vector]EXAMPLES:

sage: S = sandlib('generic') sage: c = S.max_stable() + S.identity() sage: c.stabilize(True) [{1: 2, 2: 2, 3: 1, 4: 1, 5: 1}, {1: 1, 2: 5, 3: 7, 4: 1, 5: 6}] sage: S.max_stable() & S.identity() {1: 2, 2: 2, 3: 1, 4: 1, 5: 1} sage: S.max_stable() & S.identity() == c.stabilize() True sage: ~c {1: 2, 2: 2, 3: 1, 4: 1, 5: 1}

—

**support()**

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

INPUT:

None

OUTPUT:

list - support of the config

EXAMPLES:

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

—

**unstable()**

List of the unstable vertices.

INPUT:

None

OUTPUT:

list of vertices

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: c = SandpileConfig(S, [1,2,3]) sage: c.unstable() [2, 3]

—

**values()**

The values of the configuration as a list, sorted in the order of the vertices.

INPUT:

None

OUTPUT:

list of integers

boolean

EXAMPLES:

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

—

**Summary of methods.**

*+*— Addition of divisors.*greater-equal*—`True`if every component of`self`is at least that of`other`.*greater*—`True`if every component of`self`is at least that of`other`and the two divisors are not equal.*less-equal*—`True`if every component of`self`is at most that of`other`.*less*—`True`if every component of`self`is at most that of`other`and the two divisors are not equal.*-*— The additive inverse of the divisor.*-*— Subtraction of divisors.*add_random()*— Add one grain of sand to a random vertex.*betti()*— The Betti numbers for the simplicial complex associated with the divisor.*Dcomplex()*— The simplicial complex determined by the supports of the linearly equivalent effective divisors.*deg()*— The degree of the divisor.*dualize()*— The difference between the maximal stable divisor and the divisor.*effective_div(verbose=True)*— All linearly equivalent effective divisors.*fire_script(sigma)*— Fire the script`sigma`, i.e., fire each vertex the indicated number of times.*fire_unstable()*— Fire all unstable vertices.*fire_vertex(v)*— Fire the vertex`v`.*is_alive(cycle=False)*— Will the divisor stabilize under repeated firings of all unstable vertices?*is_symmetric(orbits)*— Is the divisor constant over the vertices in each sublist of`orbits`?*linear_system()*— The complete linear system of a divisor.*r_of_D(verbose=False)*— Returns`r(D)`.*sandpile()*— The divisor’s underlying sandpile.*show(heights=True,directed=None,kwds)*— Show the divisor.*support()*— List of keys of the nonzero values of the divisor.*unstable()*— List of the unstable vertices.*values()*— The values of the divisor as a list, sorted in the order of the vertices.

**Complete descriptions of SandpileDivisor methods.**

—

**+**

Addition of divisors.

INPUT:

other- SandpileDivisorOUTPUT:

sum of

selfandotherEXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [3,2,1]) sage: D + E {0: 4, 1: 4, 2: 4}

—

**>=**

Trueif every component ofselfis at least that ofother.INPUT:

other- SandpileDivisorOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [2,3,4]) sage: F = SandpileDivisor(S, [2,0,4]) sage: D >= D True sage: E >= D True sage: D >= E False sage: F >= D False sage: D >= F False

—

**>**

Trueif every component ofselfis at least that ofotherand the two divisors are not equal.INPUT:

other- SandpileDivisorOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [1,3,4]) sage: D > D False sage: E > D True sage: D > E False

—

**<=**

Trueif every component ofselfis at most that ofother.INPUT:

other- SandpileDivisorOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [2,3,4]) sage: F = SandpileDivisor(S, [2,0,4]) sage: D <= D True sage: D <= E True sage: E <= D False sage: D <= F False sage: F <= D False

—

**<**

Trueif every component ofselfis at most that ofotherand the two divisors are not equal.INPUT:

other- SandpileDivisorOUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [2,3,4]) sage: D < D False sage: D < E True sage: E < D False

—

**-**

The additive inverse of the divisor.

INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: -D {0: -1, 1: -2, 2: -3}

—

**-**

Subtraction of divisors.

INPUT:

other- SandpileDivisorOUTPUT:

Difference of

selfandotherEXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: E = SandpileDivisor(S, [3,2,1]) sage: D - E {0: -2, 1: 0, 2: 2}

—

**add_random()**

Add one grain of sand to a random vertex.

INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

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

—

**betti()**

The Betti numbers for the simplicial complex associated with the divisor.

INPUT:

None

OUTPUT:

dictionary of integers

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [2,0,1]) sage: D.betti() # optional - 4ti2 {0: 1, 1: 1}

—

**Dcomplex()**

The simplicial complex determined by the supports of the linearly equivalent effective divisors.

INPUT:

None

OUTPUT:

simplicial complex

EXAMPLES:

sage: S = sandlib('generic') sage: p = SandpileDivisor(S, [0,1,2,0,0,1]).Dcomplex() # optional - 4ti2 sage: p.homology() # optional - 4ti2 {0: 0, 1: Z x Z, 2: 0, 3: 0} sage: p.f_vector() # optional - 4ti2 [1, 6, 15, 9, 1] sage: p.betti() # optional - 4ti2 {0: 1, 1: 2, 2: 0, 3: 0}

—

**deg()**

The degree of the divisor.

INPUT:

None

OUTPUT:

integer

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.deg() 6

—

**dualize()**

The difference between the maximal stable divisor and the divisor.

INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) 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

—

**effective_div(verbose=True)**

All linearly equivalent effective divisors. If

verboseisFalse, the divisors are converted to lists of integers.INPUT:

verbose(optional) - booleanOUTPUT:

list (of divisors)

EXAMPLES:

sage: S = sandlib('generic') sage: D = SandpileDivisor(S, [0,0,0,0,0,2]) # optional - 4ti2 sage: D.effective_div() # optional - 4ti2 [{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: D.effective_div(False) # optional - 4ti2 [[1, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 2]]

—

**fire_script(sigma)**

Fire the script

sigma, i.e., fire each vertex the indicated number of times.INPUT:

sigma- SandpileDivisor or (list or dict representing a SandpileDivisor)OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) 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

—

**fire_unstable()**

Fire all unstable vertices.

INPUT:

None

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.fire_unstable() {0: 3, 1: 1, 2: 2}

—

**fire_vertex(v)**

Fire the vertex

v.INPUT:

v- vertexOUTPUT:

SandpileDivisor

EXAMPLES:

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

—

**is_alive(cycle=False)**

Will the divisor stabilize under repeated firings of all unstable vertices? Optionally returns the resulting cycle.

INPUT:

cycle(optional) - booleanOUTPUT:

boolean or optionally, a list of SandpileDivisors

EXAMPLES:

sage: S = complete_sandpile(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}]

—

**is_symmetric(orbits)**

This function checks if the values of the divisor are constant over the vertices in each sublist of

orbits.INPUT:

orbits- 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: D = SandpileDivisor(S, [2,1, 2, 2, 3]) sage: D.is_symmetric([[0,2,3]]) True

—

**linear_system()**

The complete linear system of a divisor.

INPUT: None

OUTPUT:

dict -

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

sage: S = sandlib('generic') sage: D = SandpileDivisor(S, [0,0,0,0,0,2]) sage: D.linear_system() # optional - 4ti2 {'inhomog': [[0, 0, -1, -1, 0, -2], [0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0]], 'num_inhomog': 3, 'num_homog': 2, 'homog': [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0]]}NOTES:

If L is the Laplacian, an arbitrary v such that v * L>= -D has the form v = w + t where w is in

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

This method requires 4ti2.

—

**r_of_D(verbose=False)**

Returns

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

verbose(optional) - booleanOUTPUT:

integer

r(D)or tuple (integerr(D), divisorF)EXAMPLES:

sage: S = sandlib('generic') sage: D = SandpileDivisor(S, [0,0,0,0,0,4]) # optional - 4ti2 sage: E = D.r_of_D(True) # optional - 4ti2 sage: E # optional - 4ti2 (1, {0: 0, 1: 1, 2: 0, 3: 1, 4: 0, 5: 0}) sage: F = E[1] # optional - 4ti2 sage: (D - F).values() # optional - 4ti2 [0, -1, 0, -1, 0, 4] sage: (D - F).effective_div() # optional - 4ti2 [] sage: SandpileDivisor(S, [0,0,0,0,0,-4]).r_of_D(True) # optional - 4ti2 (-1, {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: -4})

—

**sandpile()**

The divisor’s underlying sandpile.

INPUT:

None

OUTPUT:

Sandpile

EXAMPLES:

sage: S = sandlib('genus2') sage: D = SandpileDivisor(S,[1,-2,0,3]) sage: D.sandpile() Digraph on 4 vertices sage: D.sandpile() == S True

—

**show(heights=True,directed=None,kwds)**

Show the divisor.

INPUT:

heights- whether to label each vertex with the amount of sandkwds- arguments passed to the show method for Graphdirected- whether to draw directed edgesOUTPUT:

None

EXAMPLES:

sage: S = sandlib('genus2') sage: D = SandpileDivisor(S,[1,-2,0,2]) sage: D.show(graph_border=True,vertex_size=700,directed=False)

—

**support()**

List of keys of the nonzero values of the divisor.

INPUT:

None

OUTPUT:

list - support of the divisor

EXAMPLES:

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

—

**unstable()**

List of the unstable vertices.

INPUT:

None

OUTPUT:

list of vertices

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.unstable() [1, 2]

—

**values()**

The values of the divisor as a list, sorted in the order of the vertices.

INPUT:

None

OUTPUT:

list of integers

boolean

EXAMPLES:

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

*admissible_partitions(S, k)*— Partitions of the vertices into`k`parts, each of which is connected.*aztec_sandpile(n)*— The aztec diamond graph.*complete_sandpile(n)*— Sandpile on the complete graph.*firing_graph(S, eff)*— The firing graph.*firing_vector(S,D,E)*— The firing vector taking divisor`D`to divisor`E`.*glue_graphs(g,h,glue_g,glue_h)*— Glue two sandpiles together.*grid_sandpile(m,n)*— The \(m\times n\) grid sandpile.*min_cycles(G,v)*— The minimal length cycles in the digraph`G`starting at vertex`v`.*parallel_firing_graph(S,eff)*— The parallel-firing graph.*partition_sandpile(S,p)*— Sandpile formed with vertices consisting of parts of an admissible partition.*random_digraph(num_verts,p=1/2,directed=True,weight_max=1)*— A random directed graph.*random_DAG(num_verts,p=1/2,weight_max=1)*— A random directed acyclic graph.*random_tree(n,d)*— Random tree sandpile.*sandlib(selector=None)*— A collection of sandpiles.*triangle_sandpile(n)*— The triangle sandpile.*wilmes_algorithm(M)*— Find matrix with the same integer row span as`M`that is the reduced Laplacian of a digraph.

**Complete descriptions of methods.**

**admissible_partitions(S, k)**

The partitions of the vertices of

Sintokparts, each of which is connected.INPUT:

S- Sandpilek- integerOUTPUT:

list of partitions

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: P = [admissible_partitions(S, i) for i in [2,3,4]] sage: P [[{{0}, {1, 2, 3}}, {{0, 2, 3}, {1}}, {{0, 1, 3}, {2}}, {{0, 1, 2}, {3}}, {{0, 1}, {2, 3}}, {{0, 3}, {1, 2}}], [{{0}, {1}, {2, 3}}, {{0}, {1, 2}, {3}}, {{0, 3}, {1}, {2}}, {{0, 1}, {2}, {3}}], [{{0}, {1}, {2}, {3}}]] sage: for p in P: # long time ....: sum([partition_sandpile(S, i).betti(verbose=false)[-1] for i in p]) # long time 6 8 3 sage: S.betti() # long time 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 6 8 3 ------------------------------ total: 1 6 8 3

—

**aztec(n)**

The aztec diamond graph.

INPUT:

n - integer

OUTPUT:

dictionary for the aztec diamond graph

EXAMPLES:

sage: aztec_sandpile(2) {'sink': {(-3/2, -1/2): 2, (-3/2, 1/2): 2, (-1/2, -3/2): 2, (-1/2, 3/2): 2, (1/2, -3/2): 2, (1/2, 3/2): 2, (3/2, -1/2): 2, (3/2, 1/2): 2}, (-3/2, -1/2): {'sink': 2, (-3/2, 1/2): 1, (-1/2, -1/2): 1}, (-3/2, 1/2): {'sink': 2, (-3/2, -1/2): 1, (-1/2, 1/2): 1}, (-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): {'sink': 2, (-1/2, 1/2): 1, (1/2, 3/2): 1}, (1/2, -3/2): {'sink': 2, (-1/2, -3/2): 1, (1/2, -1/2): 1}, (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): {'sink': 2, (-1/2, 3/2): 1, (1/2, 1/2): 1}, (3/2, -1/2): {'sink': 2, (1/2, -1/2): 1, (3/2, 1/2): 1}, (3/2, 1/2): {'sink': 2, (1/2, 1/2): 1, (3/2, -1/2): 1}} sage: Sandpile(aztec_sandpile(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.

—

**complete_sandpile(n)**

The sandpile on the complete graph with n vertices.

INPUT:

n- positive integerOUTPUT:

Sandpile

EXAMPLES:

sage: K = complete_sandpile(5) sage: K.betti(verbose=False) # long time [1, 15, 50, 60, 24]

—

**firing_graph(S, eff)**

Creates a digraph with divisors as vertices and edges between two divisors

DandEif firing a single vertex inDgivesE.INPUT:

S- sandpileeff- list of divisorsOUTPUT:

DiGraph

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(6),0) sage: D = SandpileDivisor(S, [1,1,1,1,2,0]) sage: eff = D.effective_div() # optional - 4ti2 sage: firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01) # optional 4ti2

—

**firing_vector(S,D,E)**

If

DandEare linearly equivalent divisors, find the firing vector takingDtoE.INPUT:

S-SandpileD,E- tuples (representing linearly equivalent divisors)OUTPUT:

tuple (representing a firing vector from

DtoE)EXAMPLES:

sage: S = complete_sandpile(4) sage: D = SandpileDivisor(S, {0: 0, 1: 0, 2: 8, 3: 0}) sage: E = SandpileDivisor(S, {0: 2, 1: 2, 2: 2, 3: 2}) sage: v = firing_vector(S, D, E) sage: v (0, 0, 2, 0) sage: vector(D.values()) - S.laplacian()*vector(v) == vector(E.values()) TrueThe divisors must be linearly equivalent:

sage: S = complete_sandpile(4) sage: D = SandpileDivisor(S, {0: 0, 1: 0, 2: 8, 3: 0}) sage: firing_vector(S, D, S.zero_div()) Error. Are the divisors linearly equivalent?

—

**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.h_vector() [1, 6, 17, 31, 41, 41, 31, 17, 6, 1] sage: S.resolution() # long time 'R^1 <-- 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 \(0\).Both

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

—

**grid_sandpile(m,n)**

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

INPUT:

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

sink.EXAMPLES:

sage: grid_sandpile(3,4).dict() {'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: grid_sandpile(3,4).group_order() 4140081

—

**min_cycles(G,v)**

Minimal length cycles in the digraph

Gstarting at vertexv.INPUT:

G- DiGraphv- vertex ofGOUTPUT:

list of lists of vertices

EXAMPLES:

sage: T = sandlib('gor') sage: [min_cycles(T, i) for i in T.vertices()] [[], [[1, 3]], [[2, 3, 1], [2, 3]], [[3, 1], [3, 2]]]

—

**parallel_firing_graph(S,eff)**

Creates a digraph with divisors as vertices and edges between two divisors

DandEif firing all unstable vertices inDgivesE.INPUT:

S- Sandpileeff- list of divisorsOUTPUT:

DiGraph

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(6),0) sage: D = SandpileDivisor(S, [1,1,1,1,2,0]) sage: eff = D.effective_div() # optional - 4ti2 sage: parallel_firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01) # optional - 4ti2

—

**partition_sandpile(S,p)**

Each set of vertices in

pis regarded as a single vertex, with and edge betweenAandBif some element ofAis connected by an edge to some element ofBinS.INPUT:

S- Sandpilep- partition of the vertices ofSOUTPUT:

Sandpile

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(4), 0) sage: P = [admissible_partitions(S, i) for i in [2,3,4]] sage: for p in P: #long time ....: sum([partition_sandpile(S, i).betti(verbose=false)[-1] for i in p]) # long time 6 8 3 sage: S.betti() # long time 0 1 2 3 ------------------------------ 0: 1 - - - 1: - 6 8 3 ------------------------------ total: 1 6 8 3

—

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

A random weighted digraph with a directed spanning tree rooted at \(0\). If

directed = False, the only difference is that if \((i,j,w)\) is an edge with tail \(i\), head \(j\), and weight \(w\), then \((j,i,w)\) 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_digraph(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 \(0\) and \(1\)weight_max– integer greater than \(0\)OUTPUT:

directed acyclic graph with sink \(0\)

EXAMPLES:

sage: S = random_DAG(5, 0.3)

—

**random_tree(n,d)**

Returns a random undirected tree with

nnodes, no node having degree higher thand.INPUT:

n,d- integersOUTPUT:

Graph

EXAMPLES:

sage: T = random_tree(15,3) sage: T.show() sage: S = Sandpile(T,0) sage: U = S.reorder_vertices() sage: Graph(U).show()

—

**sandlib(selector=None)**

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 genus2 : Undirected graph of genus 2 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

—

**triangle(n)**

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- intOUTPUT:

Sandpile

EXAMPLES:

sage: T = triangle_sandpile(5) sage: T.group_order() 135418115000

—

**wilmes_algorithm(M)**

Computes an integer matrix

Lwith the same integer row span asMand such thatLis the reduced laplacian of a directed multigraph.INPUT:

M- square integer matrix of full rankOUTPUT:

L- integer matrixEXAMPLES:

sage: P = matrix([[2,3,-7,-3],[5,2,-5,5],[8,2,5,4],[-5,-9,6,6]]) sage: wilmes_algorithm(P) [ 1642 -13 -1627 -1] [ -1 1980 -1582 -397] [ 0 -1 1650 -1649] [ 0 0 -1658 1658]NOTES:

The algorithm is due to John Wilmes.

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

```
sage: SandpileConfig.fire_vertex?
Base Class: <type 'instancemethod'>
String Form: <unbound method SandpileConfig.fire_vertex>
Namespace: Interactive
File: /usr/local/sage-4.7/local/lib/python2.6/site-packages/sage/sandpiles/sandpile.py
Definition: SandpileConfig.fire_vertex(self, v)
Docstring:
Fire the vertex ``v``.
INPUT:
``v`` - vertex
OUTPUT:
SandpileConfig
EXAMPLES:
sage: S = Sandpile(graphs.CycleGraph(3), 0)
sage: c = SandpileConfig(S, [1,2])
sage: c.fire_vertex(2)
{1: 2, 2: 0}
```

Note

An alternative to `SandpileConfig.fire_vertex?` in the preceding code example
would be `c.fire_vertex?`, if `c` is any SandpileConfig.

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] | (1, 2) 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. |

[CRS] | Robert Cori, Dominique Rossin, and Bruno Salvy, Polynomial ideals for sandpiles and their Gröbner bases, Theoretical Computer Science, 276 (2002) no. 1–2, 1–15. |

[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). |