A Shuffling Theorem for Reflectively Symmetric Tilings

TL;DR

This paper extends a shuffling theorem to reflectively symmetric tilings of doubly-dented hexagons, revealing how certain local modifications affect tiling counts and generalizing previous results in halved hexagon enumeration.

Contribution

It introduces a new shuffling theorem for reflectively symmetric tilings and generalizes known results for halved hexagons, expanding the understanding of tiling enumeration.

Findings

Shuffling along a horizontal axis changes tiling numbers by a simple factor.

Theorems are proved for reflectively symmetric tilings of doubly-dented hexagons.

Generalizations of known results for halved hexagons are established.

Abstract

In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, the reflectively symmetric tilings, of the doubly-dented hexagons. We also prove several shuffling theorems for halved hexagons. These theorems generalize a number of known results in the enumeration of halved hexagons.