Sandpiles and Dominos

We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m x 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m x 2n checkerboard on a M\"obius strip.


Introduction
This paper relates the Abelian Sandpile Model (ASM) on a grid graph to domino tilings of checkerboards. The ASM is, roughly, a game in which one places grains of sand on the vertices of a graph, Γ, whose vertices and edges we assume to be finite in number. If the amount of sand on a vertex reaches a certain threshold, the vertex becomes unstable and fires, sending a grain of sand to each of its neighbors. Some of these neighbors, in turn, may now be unstable. Thus, adding a grain of sand to the system may set off a cascade of vertex firings. The resulting "avalanche" eventually subsides, even though our graph is finite, since the system is not conservative: there is a special vertex that serves as a sink, absorbing any sand that reaches it. It is assumed that every vertex is connected to the sink by a path of edges, so as a consequence, every pile of sand placed on the graph stabilizes after a finite number of vertex firings. It turns out that this stable state only depends on the initial configuration of sand, not on the order of the firings of unstable vertices, which accounts for the use of the word "abelian." Now imagine starting with no sand on Γ then repeatedly choosing a vertex at random, adding a grain of sand, and allowing the pile of sand to stabilize. In the resulting sequence of configurations of sand, certain configurations will appear infinitely often. These are the so-called "recurrent" configurations. A basic theorem in the theory of sandpiles is that the collection of recurrent configurations forms an additive group, where addition is defined as vertex-wise addition of grains of sand, followed by stabilization. This group is called the sandpile group or critical group of Γ. Equivalent versions of the sandpile group have arisen independently. For a history and as a general reference, see [14].
In their seminal 1987 paper, Bak, Tang, and Wiesenfeld (BTW), [1], studied sandpile dynamics in the case of what we call the sandpile grid graph. To construct the m × n sandpile grid graph, start with the ordinary grid graph with vertices [m] × [n] and edges {(i, j), (i , j )} such that |i − i | + |j − j | = 1. Then add a new vertex to serve as a sink, and add edges from the boundary vertices to the sink so that each vertex on the grid has degree 4. Thus, corner vertices have two edges to the sink (assuming m and n are greater than 1), as on the left in Figure 6. Dropping one grain of sand at a time onto a sandpile grid graph and letting the system stabilize, BTW experimentally finds that eventually the system evolves into a barely stable "self-organized critical" state. This critical state is characterized by the property that the sizes of avalanches caused by dropping a single grainmeasured either temporally (by the number of ensuing vertex firings) or spatially (by the number of different vertices that fire)-obey a power law. The power-laws observed by BTW in the case of some sandpile grid graphs have not yet been proven.
The ASM, due to Dhar [8], is a generalization of the BTW model to a wider class of graphs. It was Dhar who made the key observation of its abelian property and who coined the term "sandpile group" for the collection of recurrent configurations of sand. In terms of the ASM, the evolution to a critical state observed by BTW comes from the fact that by repeatedly adding a grain of sand to a graph and stabilizing, one eventually reaches a configuration that is recurrent. Past this point, each configuration reached by adding sand and stabilizing is again recurrent.
The initial motivation for our work was a question posed to the second and third authors by Irena Swanson. She was looking at an online computer program [24] for visualizing the ASM on a sandpile grid graph. By pushing a button, the program adds one grain of sand to each of the nonsink vertices then stabilizes the resulting configuration. Swanson asked, "Starting with no sand, how many times would I need to push this button to get the identity of the sandpile group?" A technicality arises here: the configuration consisting of one grain of sand on each vertex is not recurrent, hence, not in the group. However, the all-2s configuration, having two grains at each vertex, is recurrent. So for the sake of this introduction, we reword the question as: "What is the order of the all-2s configuration?" Looking at data (cf. Section 5, Table 1), one is naturally led to the special case of the all-2s configuration on the 2n × 2n sandpile grid graph, which we denote by 2 2n×2n . The orders for 2 2n×2n for n = 1, . . . , 5 are 1, 3, 29, 901, 89893.
Plugging these numbers into the Online Encyclopedia of Integer Sequences yields a single match, sequence A065072 ( [29]): the sequence of odd integers (a n ) n≥1 such that 2 n a 2 n is the number of domino tilings of the 2n × 2n checkerboard. 1 (Some  background on this sequence is included in Section 5.) So we conjectured that the order of 2 2n×2n is equal to a n , and trying to prove this is what first led to the ideas presented here. Difficulty in finishing our proof of the conjecture led to further computation, at which time we (embarrassingly) found that the order of 2 2n×2n for n = 6 is, actually, 5758715 = a 6 /5. Thus, the conjecture is false, and there are apparently at least two natural sequences that start 1, 3, 29, 901, 89893! Theorem 5.5 shows that the cyclic group generated by 2 2n×2n is isomorphic to a subgroup of a sandpile group whose order is a n , and therefore the order of 2 2n×2n divides a n . We do not know when equality holds, and we have not yet answered Irena Swanson's question. On the other hand, further experimentation using the mathematical software Sage led us to a more fundamental connection between the sandpile group and domino tilings of the grid graph. The connection is due to a property that is a notable feature of the elements of the subgroup generated by the all-2s configurationsymmetry with respect to the central horizontal and vertical axes. The recurrent identity element for the sandpile grid graph, as exhibited in Figure 1, also has this symmetry. 2 If Γ is any graph equipped with an action of a finite group G, it is natural to consider the collection of G-invariant configurations. Proposition 2.6 establishes that the symmetric recurrent configurations form a subgroup of the sandpile group for Γ. The central purpose of this paper is to explain how symmetry links the sandpile group of the grid graph to domino tilings.
We now describe our main results. We study the recurrent configurations on the sandpile grid graph having Z/2 × Z/2 symmetry with respect to the central horizontal and vertical axes. The cases of even×even-, even×odd-, and odd×odddimensional grids each have their own particularities, and so we divide their analysis into separate cases, resulting in Theorems 4.2, 4.5, and 4.10, respectively. In each case, we compute the number of symmetric recurrents as (i) the number of domino tilings of corresponding (weighted) rectangular checkerboards; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions.
The double-product in equation (1.1) is an instance of the famous formula due to Kasteleyn [18] and to Temperley and Fisher [39] for the number of domino tilings of a 2m × 2n checkerboard: m h=1 n k=1 4 cos 2 hπ 2m + 1 + 4 cos 2 kπ 2n + 1 , for which Theorem 4.2 provides a new proof.
In the case of the even×odd grid, there is an extra "twist": the double-product in Theorem 4.5 for the even×odd grid is (a slight re-writing of) the formula of Lu and Wu [21] for the number of domino tilings of a checkerboard on a Möbius strip.
To sketch the main idea behind the proofs of these theorems, suppose a group G acts on a graph Γ with fixed sink vertex (cf. Section 2.2). To study symmetric configurations with respect to the action of G, one considers a new firing rule in which a vertex only fires simultaneously with all other vertices in its orbit under G. This new firing rule can be encoded in an m × m matrix D where m is the number of orbits of nonsink vertices of G. We show in Corollary 2.11 that det(D) is the number of symmetric recurrents on G. Suppose, as is the case for for sandpile grid graphs, that either D or its transpose happens to be the (reduced) Laplacian of an associated graph Γ . The nonsink vertices correspond to the orbits of vertices of the original graph. The well-known matrix-tree theorem says that the determinant of D is the number of spanning trees of Γ . Then the generalized Temperley bijection [19] says these spanning trees correspond with perfect matchings of a third graph Γ . In this way, the symmetric recurrents on Γ can be put into correspondence with the perfect matchings of Γ . In the case where Γ is a sandpile grid graph, Γ is a weighted grid graph, and perfect matchings of it correspond to weighted tilings of a checkerboard. Also, in this case, the matrix D has a nice block triangular form (cf. Lemma 4.1), which leads to a recursive formula for its determinant and a connection with Chebyshev polynomials.  5 The order of the all-twos configuration. Corollary 5.6: the order of the all-2s configuration on the 2n × 2n sandpile grid divides the odd number a n such that 2 n a 2 n is the number of domino tilings of the 2n × 2n checkerboard. 6 Conclusion. A list of open problems. of edges running from v to w. In particular, wt(v, w) > 0 if and only if (v, w) ∈ E. The vertex s is called the sink of Γ, and it is assumed that each vertex of Γ has a directed path to s. Let V := V \ {s} be the set of non-sink vertices. A (sandpile) configuration on Γ is an element of N V , the free monoid on V . If c = v∈ V c v v is a configuration, we think of each component, c v , as a number of grains of sand stacked on vertex v.
, is the out-degree of v, i.e., the number of directed edges emanating from v. If v is unstable in c, we may fire (topple) c at v to get a new configuration c defined for each w ∈ V by In other words, If the configurationc is obtained from c by a sequence of firings of unstable vertices, we write c →c.
Since each vertex has a path to the sink, s, it turns out that by repeatedly firing unstable vertices each configuration relaxes to a stable configuration. Moreover, this stable configuration is independent of the ordering of firing of unstable vertices. Thus, we may talk about the stabilization of a configuration c, which we denote by c • . Define the binary operation of stable addition on the set of all configurations as component-wise addition followed by stabilization. In other words, the stable addition of configurations a and b is given by Let M denote the collection of stable configurations on Γ. Then stable addition restricted to M makes M into a commutative monoid. A configuration c on Γ is recurrent if: (1) it is stable, and (2) given any configuration a, there is a configuration b such that (a + b) • = c. The maximal stable configuration, c max , is defined by It turns out that the collection of recurrent configurations forms a principal semiideal of M generated by c max . This means that the recurrent configurations are exactly those obtained by adding sand to the maximal stable configuration and stabilizing. Further, the collection of recurrent configurations forms a group, S(Γ), called the sandpile group for Γ. Note that the identity for S(Γ) is not usually the zero-configuration, 0 ∈ N V .
For an undirected graph, i.e., a graph for which wt(u, v) = wt(v, u) for each pair of vertices u and v, one may use the burning algorithm, due to Dhar [9], to determine whether a configuration is recurrent (for a generalization to directed graphs, see [37]): Then in the stabilization of b + c, each vertex fires at most once, and the following are equivalent: (1) c is recurrent; (3) in the stabilization of b + c, each non-sink vertex fires.
Define the proper Laplacian, L : for each function f ∈ Z V . Taking the Z-dual (applying the functor Hom( · , Z)) gives the mapping of free abelian groups We call ∆ the Laplacian of Γ. Restricting ∆ to Z V and setting the component of s equal to 0 gives the reduced Laplacian, ∆ : There is a well-known isomorphism While there may be many stable configurations in each equivalence class of Z V modulo image( ∆), there is only one that is recurrent. For instance, the recurrent element in the equivalence class of 0 is the identity of S(Γ).
A spanning tree of Γ rooted at s is a directed subgraph containing all the vertices, having no directed cycles, and for which s has no out-going edges while every other vertex has exactly one out-going edge. The weights of the edges of a spanning tree are the same as they are for Γ, and the weight of a spanning tree is the product of the weights of its edges. The matrix-tree theorem says the sum of the weights of the set of all spanning trees of Γ rooted at s is equal to det ∆, the determinant of the reduced Laplacian. It then follows from (2.1) that the number of elements of the sandpile group is also the sum of the weights of the spanning trees rooted at s.

Symmetric configurations.
Preliminary versions of the results in this section appear in [11]. Let G be a finite group. An action of G on Γ is an action of G on V fixing s, sending edges to edges, and preserving edge-weights. In detail, it is a mapping (4) if (v, w) ∈ E, then (gv, gw) ∈ E and both edges have the same weight. Note that these conditions imply that outdeg(v) = outdeg(gv) for all v ∈ V and g ∈ G. For the rest of this section, let G be a group acting on Γ.
By linearity, the action of G extends to an action on NV and ZV . Since G fixes the sink, G acts on configurations and each element of G induces an automorphism of S(Γ) (cf. 2.3). We say a configuration c is symmetric (with respect to the action by G) if gc = c for all g ∈ G.
Proposition 2.2. The action of G commutes with stabilization. That is, if c is any configuration on Γ and g ∈ G, then g(c • ) = (gc) • .
Proof. Suppose that c is stabilized by firing the sequence of vertices v 1 , . . . , v t . Then At the k-th step in the stabilization process, c has relaxed to the configuration Thus, we can fire the sequence of vertices gv 1 , . . . , gv t in gc, resulting in the stable configuration Corollary 2.3. The action of G preserves recurrent configurations, i.e., if c is a recurrent configuration and g ∈ G, then gc is recurrent.
Proof. If c is recurrent, we can find a configuration b such that c = (b + c max ) • . Then, Corollary 2.4. If c is a symmetric configuration, then so is its stabilization.
Remark 2.5. In fact, if c is a symmetric configuration, one may find a sequence of symmetric configurations, This follows since in a symmetric configuration a vertex v is unstable if and only if gv is unstable for all g ∈ G. To construct c i+1 from c i , simultaneously fire all unstable vertices of c i (an alternative is to pick any vertex v, unstable in c i , and simultaneously fire the vertices in {gv : g ∈ G}). Proposition 2.6. The collection of symmetric recurrent configurations forms a subgroup of the sandpile group S(Γ).
Proof. Since the group action respects addition in N V and stabilization, the sum of two symmetric recurrent configurations is again symmetric and recurrent. There is at least one symmetric recurrent configuration, namely, c max . Since the sandpile group is finite, it follows that these configurations form a subgroup.
Notation 2.7. The subgroup of symmetric recurrent configurations on Γ with respect to the action of the group G is denoted S(Γ) G . Proposition 2.8. If c is symmetric and recurrent then c = (a + c max ) • for some symmetric configuration a.
Proof. By [37] there exists an element b in the image of ∆ such that: one may find such a b by applying ∆ to the vector whose components are all 1s).
Then b G is symmetric and equal to zero modulo the image of ∆. Take a large positive integer N and consider N b G , the vertex-wise addition of b G with itself N times without stabilizing. Every vertex of Γ is connected by a path from a vertex in the support of b, and hence, the same is true of N b G . Thus, by choosing N large enough and by firing symmetric vertices of N b G , we obtain a symmetric Let O = O(Γ, G) = {Gv : v ∈ V } denote the set of orbits of the non-sink vertices.
The symmetrized reduced Laplacian is the Z-linear mapping

thus obtaining a bijection between symmetric elements of
the symmetric configuration obtained from c by firing all vertices in the orbit of v.
For the following let r : Z V /image( ∆) → S(Γ) denote the inverse of the isomorphism in (2.1).
Proposition 2.10. There is an isomorphism of groups, To see that the image of r • Λ is symmetric, consider the symmetric configuration and consider the isomorphism g : Z V → Z V determined by the action of g on vertices. A straightforward calculation shows that ∆ = g ∆g −1 . It follows that Corollary 2.11. The number of symmetric recurrent configurations is Remark 2.12. We have not assumed that the action of G on Γ is faithful. If K is the kernel of the action of G, then O(Γ, G) = O(Γ, G/K) and S G = S G/K . We also have ∆ G = ∆ G/K . Let G = {e, g} be the group of order 2 with identity e. Consider the action of G on Γ for which g swaps vertices u and v and fixes vertices w and s. Ordering the vertices of Γ as u, v, w and ordering the orbits, O, as Gu, Gw, the reduced Laplacian and the symmetrized reduced Laplacian for Γ become

Gu Gw
where we have labeled the columns by their corresponding vertices or orbits for convenience. To illustrate how one would compute the columns of the symmetrized reduced Laplacian in general, consider the column of ∆ G corresponding to Gu = {u, v}. It was computed by first adding the u-and v-columns of ∆ to get the 3vector = (2, 2, −2), then taking the u and w components of since u and w were chosen as orbit representatives.

Matchings and trees
In this section, assume that Γ = (V, E, wt, s) is embedded in the plane, and fix a face f s containing the sink vertex, s. In §4 and §5, we always take f s to be the unbounded face. We recall the generalized Temperley bijection, due to [19], between directed spanning trees of Γ rooted at s and perfect matchings of a related weighted undirected graph, H(Γ). (The graph H(Γ) would be denoted H(s, f s ) in [19].) It is sometimes convenient to allow an edge e = (u, v) to be represented in the embedding by distinct weighted edges e 1 , . . . , e k , each with tail u and head v, such that k i=1 wt(e i ) = wt(e). Also, we would like to be able to embed a pair of oppositely oriented edges between the same vertices so that they coincide in the plane. For these purposes then, we work in the more general category of weighted directed multi-graphs by allowing E to be a multiset of edges in which an edge e with endpoints u and v is represented as the set e = {u, v} with a pair of weights wt(e, (u, v)) and wt(e, (v, u)), at least one of which is nonzero. Each edge in the embedding is then represented by a double-headed arrow with two weight labels (the label wt(e, (u, v)) being placed next to the head vertex, v). Figure 3 shows a pair of edges e = {u, v} and e = {u, v} where wt(e, (u, v))) = 2, wt(e, (v, u))) = 0, wt(e , (u, v))) = 3, and wt(e , (v, u))) = 1. The top edge, e, represents a single directed edge (u, v) of weight 2, and the bottom edge represents a pair of directed edges of weights 3 and 1. The two edges combine to represent a pair of directed edges, (u, v) of weight 5 and (v, u) of weight 1. The rough idea of the construction of the weighted undirected graph H(Γ) is to overlay the embedded graph Γ with its dual, forgetting the orientation of the edges and introducing new vertices where their edges cross. Then remove s and the vertex corresponding to the chosen face f s , and remove their incident edges. In detail, the vertices of H(Γ) are where F is the set of faces of Γ, including the unbounded face, and the edges of H(Γ) are The weight of each edge of the form {t u , t e } with e = {u, v} ∈ E is defined to be wt(e, (u, v)), and the weight of each edge of the form {t e , t f } with f ∈ F is defined to be 1. Figure 4 depicts a graph Γ embedded in the plane (for which the multiset E is actually just a set). The graph displayed in the middle is the superposition of Γ with its dual, Γ ⊥ . The unbounded face is chosen as f s . For convenience, its corresponding vertex is omitted from the middle graph, and its incident edges are only partially drawn.  A perfect matching of a weighted undirected graph is a subset of its edges such that each vertex of the graph is incident with exactly one edge in the subset. The weight of a perfect matching is the product of the weights of its edges.
We now describe the weight-preserving bijection between perfect matchings of H(Γ) and directed spanning trees of Γ rooted at s due to [19]. Let T be a directed spanning tree of Γ rooted at s, and let T be the corresponding directed spanning tree of Γ ⊥ , the dual of Γ, rooted at f s . (The tree T is obtained by properly orienting the edges of Γ ⊥ that do not cross edges of T in Γ ∪ Γ ⊥ .) The perfect matching of H(Γ) corresponding to T consists of the following: (1) an edge {t u , t e } of weight wt(e) for each e = (u, v) ∈ T ; (2) an edge {t f , t e } of weight 1 for eachẽ = (f, f ) ∈ T , where e is the edge in Γ that crossed byẽ. See Figure 5 for an example continuing the example from Figure 4. As discussed in [19], although H(Γ) depends on the embedding of Γ and on the choice of f s , the number of spanning trees of Γ rooted at s (and hence, the number of perfect matchings of H(Γ)), counted according to weight, does not change. In what follows, we will always choose f s to be the unbounded face.

Symmetric recurrents on the sandpile grid graph
The ordinary m×n grid graph is the undirected graph Γ m×n with vertices [m]×[n] and edges {(i, j), (i , j )} such that |i − i | + |j − j | = 1. The m × n sandpile grid graph, SΓ m×n , is formed from Γ m×n by adding a (disjoint) sink vertex, s, then edges incident to s so that every non-sink vertex of the resulting graph has degree 4. For instance, each of the four corners of the sandpile grid graph shares an edge of weight 2 with s in the case where m ≥ 2 and n ≥ 2, as on the left in Figure 6.
We embed Γ m×n in the plane as the standard grid with vertices arranged as in a matrix, with (1, 1) in the upper left and (m, n) in the lower right. We embed SΓ m×n similarly, but usually identify the sink vertex, s, with the unbounded face of Γ m×n for convenience in drawing, as on the left-hand side in Figure 6. The edges leading to the sink are sometimes entirely omitted from the drawing, as in Figure 10.
Our main goal in this section is to study the symmetric recurrent configurations on the sandpile grid graph. After collecting some basic facts about certain tridiagonal matrices, we divide the study into three cases: even×even-, even×odd-, and and Chebyshev polynomials of the second kind are defined by Two references are [25] and [40].
It follows from the recurrences that these polynomials may be expressed as determinants of j × j tridiagonal matrices: We have the well-known factorizations: We will also use the following well-known identities: Corollary 2.11 will be used to count the symmetric recurrents on sandpile grid graphs. The form of the determinant that arises is treated by the following. By (4.2) and (4.8), S j = U j ( 1 2 A). Hence, as required.

4.2.
Symmetric recurrents on a 2m×2n sandpile grid graph. A checkerboard is a rectangular array of squares. A domino is a 1 × 2 or 2 × 1 array of squares and, thus, covers exactly two adjacent squares of the checkerboard. A domino tiling of the checkerboard consists of placing non-overlapping dominos on the checkerboard, covering every square. As is usually done, and exhibited in Figure 7, we identify domino tilings of an m × n checkerboard with perfect matchings of Γ m×n . Figure    Part (4) of the following theorem is the well-known formula due to Kasteleyn [18] and to Temperley and Fisher [39] for the number of domino tilings of a checkerboard. We provide a new proof. (1) the number of symmetric recurrents on SΓ 2m×2n ; (2) the number of domino tilings of a 2m × 2n checkerboard; Proof. It may be helpful to read Example 4.4 in parallel with this proof. Let A n = (a h,k ) be the n × n tridiagonal matrix with entries In particular, A 1 = [3]. Take the vertices [m] × [n] as representatives for the orbits of G acting the non-sink vertices of SΓ 2m×2n . Ordering these representatives lexicographically, i.e., left-to-right then top-to-bottom, the symmetrized reduced Laplacian (2.2) is given by the mn × mn tridiagonal block matrix where I n is the n × n identity matrix and B n := A n − I n . If m = 1, then ∆ G := B n .
[(1) = (2)]: The matrix ∆ G is the reduced Laplacian of a sandpile graph we now describe. Let D m×n be the graph obtained from Γ m×n , the ordinary grid graph, by adding (i) a sink vertex, s , (ii) an edge of weight 2 from the vertex (1, 1) to s , and (iii) edges of weight 1 from each of the other vertices along the left and top sides to s , i.e., {(h, 1), s } for 1 < h ≤ m and {(1, k), s } for 1 < k ≤ n. We embed D m×n in the plane so that the non-sink vertices form an ordinary grid, and the edge of weight 2 is represented by a pair of edges of weight 1, forming a digon. Then, H(D m×n ) = Γ 2m×2n (see Figure 11). Since ∆ G = ∆ Dm×n , taking determinants shows that the number of symmetric recurrents on SΓ 2m×2n is equal to the size of the sandpile group of D m×n , and hence to the number of spanning trees of D m×n rooted at s , counted according to weight. These spanning trees are, in turn, in bijection with the perfect matchings of the graph H(D m×n ) = Γ 2m×2n obtained from the generalized Temperley bijection of Section 3. Hence, the numbers in parts (1) and (2) are equal.
[(1) = (3)]: By Corollary 2.11, det ∆ G is the number of symmetric recurrents on SΓ 2m×2n . By Lemma 4.1, Using (4.4) and the fact that the Chebyshev polynomials of the second kind satisfy Define χ 0 (x) := 1. Expanding the determinant defining χ n (x), starting along the first row, leads to a recursive formula for χ n (x): On the other hand, defining C j (x) := (−1) j U 2j (x), it follows from (4.2) that The result now follows by letting x = t h,m in (4.12), letting x = i ξ m−h,m in (4.13), and using the fact that (4.14) t h,m = 2 − 4 ξ 2 m−h,m .
[ (3) Figure 10 is colored blue. If this vertex is fired simultaneously with the other vertices in its orbit, it will lose 4 grains of sand to its neighbors but gain 1 grain of sand from the adjacent vertex in its orbit. This firing-rule is encoded in the sixth column of ∆ G 8×6 (shaded blue). Figure 10. A sandpile grid graph and its symmetrized reduced Laplacian.
The matrix ∆ G 8×6 is the reduced Laplacian of the graph D 4×3 , shown in Figure 11. To form H(D 4×3 ) = Γ 8×6 , we first overlay D 4×3 with its dual, as shown, then remove the vertices s ands and their incident edges. Figure 12 shows how a spanning tree of D 4×3 (in black) determines a spanning tree of the dual graph (in blue) and a domino tiling of the 8 × 6 checkerboard.  As part of Theorem 4.5, we will show that the domino tilings of a 2m × 2n Möbius checkerboard can be counted using weighted domino tilings of an associated ordinary checkerboard, which we now describe. Define the Möbius-weighted m × n grid graph, MΓ m×n , as the ordinary m×n grid graph but with each edge of the form checkerboard is the ordinary m × n checkerboard but for which the weight of a domino tiling is taken to be the weight of the corresponding perfect matching of MΓ m×n . In Figure 8, the dominos corresponding to edges of weight 2 are shaded. Thus, the first three tilings in the first row of Figure 8 have weights 4, 2, and 1, respectively. Example 4.9 considers a case for which m is odd. is another way to express the numbers in parts (1)-(5).
Proof of Theorem 4.5. The proof is similar to that of Theorem 4.2 after altering the definitions of the matrices A n and B n used there. This time, for n > 1, let A n = (a h,k ) be the n × n tridiagonal matrix with entries if |h − k| = 1 and h = n, −2 if h = n and k = n − 1, 0 if |h − k| ≥ 2.
[(1) = (2)]: Reasoning as in the proof of Theorem 4.2, equation (4.9) with A n and B n substituted for A n and B n gives the symmetrized reduced Laplacian, ∆ G , of SΓ 2m×(2n−1) . Unless n = 1, the matrix ∆ G is not the reduced Laplacian matrix of a sandpile graph since the sum of the elements in its penultimate column is −1 whereas the sum of the elements in any column of the reduced Laplacian of a sandpile graph must be nonnegative. However, in any case, the transpose ( ∆ G ) t is the reduced Laplacian of a sandpile graph, which we call D m×n . We embed it in the plane as a grid as we did previously with D m×n in the proof of Theorem 4.2, but this time with some edge-weights not equal to 1. Figure 15 shows D 4×3 . It is the same as D 4×3 as depicted in Figure 11, except that arrowed edges, , have been substituted for certain edges. Each represents a pair of arrows-one from right-to-left of weight 2 and one from left-to-right of weight 1-embedded so that they coincide, as discussed in Section 3. s D 4×3 Figure 15. The symmetrized reduced Laplacian for SΓ 8×5 is the reduced Laplacian for D 4×3 . Arrowed edges each represent a pair of directed edges of weights 1 and 2, respectively, as indicated by the number of arrow heads. All other edges have weight 1.
Reasoning as in the proof of Theorem 4.2, we see that the number of perfect matchings of H(D m×n ) is equal to the number of perfect matchings of MΓ 2m×(2n−1) , each counted according to weight. This number is det( ∆ G ) t = det ∆ G , which is the number of symmetric recurrents on SΓ 2m×(2n−1) by Corollary 2.11.
[(1) = (3)]: Exactly the same argument as given in the proof of Theorem 4.2 shows that where t h,m is as before, but now χ n (x) is the characteristic polymonial of A n . In light of Remark 4.6, it suffices to show χ n (t h,m ) = 2 T n (1 + 2 ξ 2 m−h,m ) for each h ∈ {0, 1, · · · , m − 1} , which we now do as before, by showing both sides of the equation satisfy the same recurrence.
Then, using identities from Section 4.1, Thus, where χ n is the characteristic polynomial of A n . Now consider the recurrences (4.15) and (4.16) in the proof of Theorem 4.5. Substituting 2s h,m for x in the former and 2 − s h,m for x in the latter, the two recurrences become the same. It follows that χ n (2 s h,m ) = 2 T n (2 − s h,m ). Then using a double-angle formula for cosine and identity (4.5),   (3) and (4) of Theorem 4.10, one may replace each ζ h,n with sin((2h − 1)π/(4n)) or, as discussed at the end of the proof of Theorem 4.5, with sin((4h − 1)π/(4n)).

The order of the all-twos configuration
Let c be a configuration on a sandpile graph Γ, not necessarily an element of S(Γ), the sandpile group. If k is a nonnegative integer, let k · c denote the vertexwise addition of c with itself k times, without stabilizing. The order of c, denoted order(c), is the smallest positive integer k such that k · c is in the image of the reduced Laplacian of Γ. If c is recurrent, then the order of c is the same as its order as an element of S(Γ) according to the isomorphism (2.1).
Consider the sandpile grid graph, SΓ m×n , with m, n ≥ 2. For each nonnegative integer k, let k m×n = k · 1 m×n be the all-ks configuration on SΓ m×n consisting of k grains of sand on each vertex. The motivating question for this section is: what is the order of 1 m×n ? Since 1 m×n has up-down and left-right symmetry, its order must divide the order of the group of symmetric recurrents on SΓ m×n calculated in Theorems 4.2, 4.5, and 4.10. The number of domino tilings of a 2n×2n checkerboard can be written as 2 n a 2 n where a n is an odd integer (cf. Proposition 5.3). Our main result is Theorem 5.5 which, through Corollary 5.6, says that the order of 2 2n×2n divides a n . (1) The configuration 1 m×n is not recurrent.
(3) The order of 1 m×n is either order( 2 m×n ) or 2 order( 2 m×n ). (4) Let ∆ m×n be the reduced Laplacian of SΓ m×n . The order of 1 m×n is the smallest integer k such that k · ∆ −1 m×n 1 m×n is an integer vector. Proof. Part (1) follows immediately from the burning algorithm (Theorem 2.1). For part (2), we start by orienting some of the edges of SΓ m×n as shown in Figure 19. First, orient all the edges containing the sink, s, so that they point away from s. Figure 19. Partial orientation of SΓ 4×5 . Arrows pointing into the grid from the outside represent edges from the sink vertex.
Next, orient all the horizontal edges to point to the right except for the last column of horizontal arrows. Finally, orient all the vertical edges down except for the last row of vertical arrows. More formally, define the partial orientation of SΓ m×n , Use O to define a poset P on the vertices of SΓ m×n by first setting u < P v if (u, v) ∈ O, then taking the transitive closure. Now list the vertices of SΓ m×n in any order v 1 , v 2 , . . . such that v i < P v j implies i < j. Thus, v 1 = s and v 2 , v 3 , v 4 , v 5 are the four corners of the grid, in some order. Starting from 2 m×n , fire v 1 . This has the effect of adding the burning configuration to 2 m×n . Since the indegree of each non-sink vertex with respect to O is 2, after v 1 , . . . , v i−1 have fired, v i is unstable. Thus, after firing the sink, every vertex will fire while stabilizing the resulting configuration. So 2 m×n is recurrent by the burning algorithm.
[note: One way to think about listing the vertices, as prescribed above, is as follows. Let P −1 := {s}, and for i ≥ 0, let P i be those elements whose distance from some corner vertex is i. (By distance from a corner vertex, we mean the length of a longest chain in P or the length of any path in O starting from a corner vertex.) For instance, P 0 consists of the four corners. After firing the vertices in P −1 , P 0 , . . . , P i−1 , all of the vertices in P i are unstable and can be fired in any order.] For part (3), let α = order( 1 m×n ) and β = order( 2 m×n ), and let e be the identity of S(SΓ m×n ). Let L denote the image of the reduced Laplacian, ∆, of SΓ m×n . Since e = (2α · 1 m×n ) • = (α · 2 m×n ) • and e = (β · 2 m×n ) • = (2β · 1 m×n ) • , we have We have (2β − α) · 1 m×n = 0 mod L. Suppose α = 2β. It cannot be that 2β − α = 1. Otherwise, 1 m×n = 0 mod L. It would then follow that 2 m×n and 3 m×n are recurrent elements equivalent to 0 modulo L, whence, 2 m×n = 3 m×n = e, a contradiction. Thus, (2β − α) · 1 m×n ≥ 2 m×n . Since 2 m×n is recurrent, ((2β − α) · 1 m×n ) • is recurrent and equivalent to 0 modulo L, and thus must be the e. So 2β − α ≥ α, and the right side of (5.1) implies α = β, as required. Now consider part (4). The order of 1 m×n is the smallest positive integer k such that k · 1 m×n = 0 mod L, i.e., for which there exists an integer vector v such that k · 1 m×n = ∆ m×n v. The result follows.
striking feature of Table 1 is the relatively small size of the numbers along the diagonal (m = n). It seems natural to group these according to parity. The sequence { 2 2n×2n } n≥1 starts 1, 3, 29, 901, 89893, . . . , which is the beginning of the famous sequence, (a n ) n≥1 , we now describe. The following was established independently by several people (cf. [17]): Proposition 5.3. The number of domino tilings of a 2n × 2n checkerboard has the form 2 n a 2 n where a n is an odd integer.
For each positive integer n, let P n be the sandpile graph with vertices V (P n ) = {v i,j : 1 ≤ i ≤ n and 1 ≤ j ≤ i} ∪ {s}.
Each v i,j is connected to those vertices v i ,j such that |i − i | + |j − j | = 1. In addition, every vertex of the form v i,n is connected to the sink vertex, s. The s s s P 1 P 2 P 3 Figure 20.
first few cases are illustrated in Figure 20. Next define a family of triangular checkerboards, H n , as in Figure 21. The checkerboard H n for n ≥ 2 is formed by adding a 2 × (2n − 1) array (width-by-height) of squares to the right of H n−1 . These graphs were introduced by M. Ciucu [5] and later used by L. Pachter [6] to Figure 21.
give the first combinatorial proof of Proposition 5.3. As part of his proof, Pachter shows that a n is the number of domino tilings of H n . As noted in [19], considering H n as a planar graph and taking its dual (forgetting about the unbounded face of H n ) gives the graph H(P n ) corresponding to P n under the generalized Temperley bijection of Section 3. See Figure 22. Proposition 5.4. The number of elements in the sandpile group for P n is # S(P n ) = a n , where a n is as in Proposition 5.3.
Proof. The number of domino tilings of H n equals the number of perfect matchings of H(P n ). By the generalized Temperley bijection, the latter is the number of spanning trees of P n , and hence, the order of the sandpile group of P n . As mentioned above, Pachter shows in [6] that a n is the number of domino tilings of H n .
The main result of this section is the following: Theorem 5.5. Let 2 2n×2n be the cyclic subgroup of S(SΓ 2n×2n ) generated by the all-2s element of Γ 2n×2n , and let 2 n denote the all-2s element on P n . Then the mapping ψ : 2 2n×2n → S(P n ), determined by ψ( 2 2n×2n ) = 2 n , is a well-defined injection of groups.
Proof. Let V n and V 2n×2n denote the non-sink vertices of P n and SΓ 2n×2n , respectively. We view configurations on P n as triangular arrays of natural numbers and configurations on SΓ 2n×2n as 2n × 2n square arrays of natural numbers. Divide the 2n × 2n grid by drawing bisecting horizontal, vertical, and diagonal lines, creating eight wedges. Define φ : Z V n → Z V 2n×2n , by placing a triangular array in the position of one of these wedges, then flipping about lines, creating a configuration on SΓ 2n×2n with dihedral symmetry. Figure 23 illustrates the case n = 4. We define special types of configurations on P n . First, let s n be the configuration in which the number of grains of sand on each vertex records that vertex's distance to the sink; then let t n denote the sandpile with no sand except for one grain on each vertex along the boundary diagonal, i.e., those vertices with degree less than 3. Figure 24 illustrates the case n = 4. Let ∆ n and ∆ 2n×2n be the reduced Laplacians for P n and SΓ 2n×2n , respectively. The following are straightforward calculations: (1) ∆ n s n = t n .
Let L n ⊂ ZV n and L 2n×2n ⊂ ZV 2n×2n denote the images of ∆ n and ∆ 2n×2n , respectively. Identify the sandpile groups of P n and SΓ 2n×2n with ZV n / L n and ZV 2n×2n / L 2n×2n , respectively. To show that ψ is well-defined and injective, we need to show that k 2 n ∈ L n for some integer k if and only if k 2 2n×2n ∈ L 2n×2n . Since the reduced Laplacians are invertible over Q, there exist unique vectors x and y defined over the rationals such that ∆ n x = 2 n and ∆ 2n×2n y = 2 2n×2n .
In other words, Using the fact that ∆ n is invertible over Q, we see that k 2 n ∈ L n if and only if kx has integer coordinates. By (5.2), this is the same as saying ky has integer components, which in turn is equivalent to k 2 2n×2n ∈ L 2n×2n , as required.
Combining this result with Proposition 5.4 gives Proposition 2.10 yields a group isomorphism between the symmetric configurations on Γ and the sandpile group S(Γ ) of Γ . By the matrix-tree theorem, the size of the latter group is the number of spanning trees of Γ (and, in fact, as mentioned earlier, SΓ is well-known to act freely and transitively on the set of spanning trees of Γ ). The generalized Temperley bijection then gives a correspondence between the spanning trees of Γ and perfect matchings of a corresponding graph, H(Γ ). Thus, the number of symmetric recurrents on Γ equals the number of perfect matchings of H(Γ ). We have applied this idea to the case of a particular group acting on sandpile grid graphs. Does it lead to anything interesting when applied to other classes of graphs with group action? The Bachelor's thesis of the first author [13] includes a discussion of the case of a dihedral action on sandpile grid graphs.