A simple explanation for the "shuffling phenomenon'' for lozenge tilings of dented hexagons

TL;DR

This paper shows that the shuffling phenomenon in lozenge tilings of dented hexagons can be explained simply through the enumeration of Gelfand--Tsetlin patterns, providing a straightforward understanding of the previously complex theorem.

Contribution

It offers a simple explanation for the shuffling phenomenon by linking it to Gelfand--Tsetlin pattern enumeration, simplifying prior proofs.

Findings

Shuffling phenomenon explained via Gelfand--Tsetlin patterns

Immediate derivation of the shuffling theorem from pattern enumeration

Connection to recent preprints on the topic

Abstract

In a recent paper, Lai and Rohatgi proved a "shuffling theorem" for lozenge tilings of a hexagon with "dents" (i.e., missing triangles). Here, we shall point out that this follows immediately from the enumeration of Gelfand--Tsetlin patterns with given bottom row. This observation is also contained in a recent preprint of Byun.