Block diagonally symmetric lozenge tilings

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.

arXiv comment19 pages, 11 figures

References in corpus (3)

• Bijective proofs of skew Schur polynomial factorizations
• Off-diagonally symmetric domino tilings of the Aztec diamond
• Off-diagonally symmetric domino tilings of the Aztec diamond of odd order