Tilings of Benzels via the Abacus Bijection

TL;DR

This paper proves conjectures about tilings of benzels, special regions in hexagonal grids, by translating the problem into square grids and utilizing a bijection between ribbon tableaux and Young tableaux.

Contribution

It resolves two of Propp's conjectures on benzels tilings and introduces a novel application of a bijection between ribbon and Young tableaux to solve tiling enumeration problems.

Findings

Resolved two conjectures on benzels tilings

Established a bijection between ribbon and Young tableaux for tiling enumeration

Extended understanding of tiling patterns in hexagonal grids

Abstract

Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and prove some additional results that he left tacit. In order to solve these problems, we first transfer benzels into the square grid. One of our primary tools, which we combine with several new ideas, is a bijection (rediscovered by Stanton and White and often attributed to them although it is considerably older) between $k$-ribbon tableaux of certain skew shapes and certain $k$-tuples of Young tableaux.