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Abstract:We introduce a new symmetry class of both plane partitions and lozenge tilings of a hexagon, called the \textit{$\mathbf{r}$-block diagonal symmetry class}, where $\mathbf{r}$ is an $n$-tuple of non-negative integers. We prove that the tiling generating function of this symmetry class under a certain weight assignment is given by a simple product formula. When $\mathbf{r}$ and weights are suitably chosen, our formula coincides with the number of domino tilings of the Aztec diamond and the number of alternating sign matrices up to a simple constant. We also deduce the volume generating function of $\mathbf{r}$-block diagonally symmetric plane partitions. Additionally, we consider \textit{$(\mathbf{r},\mathbf{r^{\prime}})$-block diagonally symmetric} lozenge tilings by embedding the hexagon into a cylinder and present an identity for the signed enumeration of this symmetry class in specific cases. Two methods are provided to study this symmetry class: (1) the method of non-intersecting lattice paths with a modification and (2) interpreting weighted lozenge tilings algebraically as (skew) Schur polynomials and applying the dual Pieri rule.

Submission history

From: Yi-Lin Lee [view email]
[v1] Tue, 25 Mar 2025 01:17:06 UTC (900 KB)
[v2] Sat, 1 Aug 2026 03:05:10 UTC (885 KB)