Domino Tilings, Domino Shuffling, and the Nabla Operator

TL;DR

The paper develops a unified framework connecting domino tilings of regions $R_\\lambda$ with nested Schröder-path families $\\mathcal{S}_\\lambda$ through the nabla operator. A key result expresses the generating polynomial $P_\\lambda(z;q,t)$ for domino tilings in terms of $\\nabla(s_\\lambda)$ via $\langle\\nabla(s_\\lambda), h_d e_{n-d}\rangle$, thereby bridging $q,t$-Catalan combinatorics to Macdonald theory. For square shapes $\\lambda=(n^n)$, the authors derive a new product formula for the Aztec diamond generating function using domino shuffling and extended ASM structures, and they establish joint symmetry of area and dinv. An open divisibility conjecture for $P_\\lambda(z;q,t)$ highlights deeper factorization patterns governed by the rank of $\\lambda$, pointing to further connections between combinatorics of tilings and symmetric functions. The work thus creates a rigorous link between tiling enumerations, path combinatorics, and Macdonald-operator structures with explicit product formulas in key cases.

Abstract

We study domino tilings of certain regions $R_λ$, indexed by partitions $λ$, 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 $λ= (n^n)$ is a square shape, domino tilings of $R_λ$ 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.

Figures (15)

• Figure 1: (a) The decomposition of $\lambda = (4,4,3,3,3,1)$ into border strips of lengths $(n_0,n_1,n_2,n_3)=(9,6,0,3)$. (b) A $\lambda$-family of Schröder paths for $\lambda = (4,4,3,3,3,1)$ with 5 diagonal steps. (c) The region $R_\lambda$ for $\lambda = (4,4,3,3,3,1)$ with the domino tiling that corresponds to the $\lambda$-family in Figure \ref{['fig:paths']}.
• Figure 2: Dinv pairs of steps from left to right corresponding to: Case 1 with $a^{(v)}_c$ undecorated, Case 1 with $a^{(v)}_c$ decorated, Case 2 with $a^{(u)}_b$ undecorated, and Case 2 with $a^{(u)}_b$ decorated.
• Figure 3: A labeled weakly-nested $(5,3,3,2)$-family of Dyck paths.
• Figure 4: Dinv pairs for any labeling with $a<b$
• Figure 5: The four types of dominoes and their corresponding path steps.

Theorems & Definitions (31)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1: LW08 and BHMPS
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm:MacdonaldIntro']}
• Theorem 3.3: Loehr