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.Comment: 35 pages, 24 figure

Abelian Sandpile Model on Symmetric Graphs

The abelian sandpile model, or chip firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sandpile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset \mathbb{R}^2$ with colored directed edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we construct a monomorphism from the sandpile group $G_{\Gamma _1}=\mathbb{Z}^{\Gamma _1}/\Delta (\mathbb{Z}^{\Gamma _1})$ on $\Gamma _1=\hat{P}_1\cap \mathbb{Z}^2$ to the one on $\Gamma _2=\hat{P}_2\cap \mathbb{Z}^2$. We provide several examples of infinite series of such tilings converging to $\mathbb{R}^2$, and thus define the limit of the sandpile group on the plane

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset \mathbb{R}^2$ with colored directed edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we construct a monomorphism from the sandpile group $G_{\Gamma _1}=\mathbb{Z}^{\Gamma _1}/\Delta (\mathbb{Z}^{\Gamma _1})$ on $\Gamma _1=\hat{P}_1\cap \mathbb{Z}^2$ to the one on $\Gamma _2=\hat{P}_2\cap \mathbb{Z}^2$. We provide several examples of infinite series of such tilings converging to $\mathbb{R}^2$, and thus define the limit of the sandpile group on the plane

Computational complexity of counting coincidences

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the problem, with $2\times 1 \times 1$ and $2\times 2 \times 1$ boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood--Richardson coefficients.Comment: 23 pages, 6 figure