# A short proof of two shuffling theorems for tilings and a weighted generalization

## TL;DR

This paper provides a unified, combinatorial proof of two shuffling theorems for lozenge tilings, extending previous results to weighted cases without using graphical condensation.

## Contribution

It introduces a new proof method based on tiling enumeration formulas, covering weighted scenarios and explaining a conjecture on symmetric tilings.

## Key findings

- Unified proof of two shuffling theorems
- Extension to weighted tilings
- Combinatorial explanation of Ciucu's conjecture

## Abstract

Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu. In this paper, we give a unified proof of these two shuffling theorems, which also covers the weighted case. Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp. Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem. Our proof also gives a combinatorial explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings.