Sandpiles and Dominos

TL;DR

This paper explores the symmetric subgroup of the abelian sandpile group on grid graphs, linking its size to domino tilings, Chebyshev polynomial values, and trigonometric sums, with new derivations for classic tiling counts.

Contribution

It introduces a novel connection between sandpile groups and domino tilings, providing new formulas and methods for counting domino tilings on various surfaces.

Findings

Size of symmetric sandpile subgroup equals domino tilings count

Derived new formulas for domino tilings of rectangular and Möbius strip grids

Connected sandpile configurations to Chebyshev polynomials and trigonometric sums

Abstract

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.