Block diagonally symmetric lozenge tilings

Abstract

We introduce a new symmetry class of both boxed plane partitions and lozenge tilings of a hexagon, called the $\mathbf{r}$-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. As a consequence, the volume generating function of $\mathbf{r}$-block symmetric plane partitions is obtained. Additionally, we consider $(\mathbf{r},\mathbf{r^{\prime}})$-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.