Domino Tilings, Domino Shuffling, and the Nabla Operator

TL;DR

This paper explores domino tilings of specific regions linked to partitions, connecting combinatorial statistics from Macdonald polynomials with algebraic operators, and provides new formulas and proofs for these tilings and their symmetries.

Contribution

It introduces a formula for domino tiling generating functions using the nabla operator and establishes a new product formula via domino shuffling, linking to alternating sign matrices.

Findings

Derived a generating polynomial formula using the nabla operator.

Established a new product formula for domino tilings of square regions.

Provided a combinatorial proof of the symmetry of area and dinv statistics.

Abstract

We study domino tilings of certain regions $R_\lambda$, indexed by partitions $\lambda$, weighted according to generalized area and dinv statistics. These statistics arise from the $q,t$-Catalan combinatorics and Macdonald polynomials. We present a formula for the generating polynomial of these domino tilings in terms of the Bergeron--Garsia nabla operator. When $\lambda = (n^n)$ is a square shape, domino tilings of $R_\lambda$ are equivalent to those of the Aztec diamond of order $n$. In this case, we give a new product formula for the resulting polynomials by domino shuffling and its connection with alternating sign matrices. In particular, we obtain a combinatorial proof of the joint symmetry of the generalized area and dinv statistics.