Lozenge tilings of hexagons with holes on three crossing lines

TL;DR

This paper presents a new, direct proof for a product formula counting lozenge tilings of hexagons with holes, extending previous results and including symmetric cases, without using graphical condensation.

Contribution

It introduces a short, direct proof for a general ratio formula of lozenge tilings of hexagons with holes, bypassing graphical condensation methods.

Findings

Derived a simple product formula for tilings of hexagons with holes.

Extended previous results to more general regions.

Provided formulas for symmetric tilings.

Abstract

The enumeration of lozenge tilings of hexagons with holes has received much attention during the last three decades. One notable feature is that a lot of the recent development involved Kuo's graphical condensation. Motivated by Ciucu, Lai and Rohatgi's work on tilings of hexagons with a removed triad of bowties, in this paper, we show that the ratio of numbers of lozenge tilings of two more general regions is expressed as a simple product formula. Our proof does not involve the graphical condensation method. The proof is short and direct. We also provide a corresponding formula for cyclically symmetric lozenge tilings. Several previous results can be easily deduced from our generalization.