Tilings of hexagons with a removed triad of bowties

TL;DR

This paper investigates lozenge tilings of hexagons with three bowtie-shaped holes on a triangular lattice, revealing a simple ratio formula for normalized regions that unifies several previous results.

Contribution

It introduces a normalized tiling ratio formula for hexagons with bowtie holes, connecting and generalizing earlier specific cases.

Findings

Derived a simple product formula for tiling ratios of normalized regions.

Unified previous results on hexagon tilings with dents and shamrocks.

Provided conceptual insight into tiling enumeration with complex holes.

Abstract

In this paper we consider arbitrary hexagons on the triangular lattice with three arbitrary bowtie-shaped holes, whose centers form an equilateral triangle. The number of lozenge tilings of such general regions is not expected --- and indeed is not --- given by a simple product formula. However, when considering a certain natural normalized counterpart of any such region, we prove that the ratio between the number of tilings of the original and the number of tilings of the normalized region is given by a simple, conceptual product formula. Several seemingly unrelated previous results from the literature --- including Lai's formula for hexagons with three dents and Ciucu and Krattenthaler's formula for hexagons with a removed shamrock --- follow as immediate consequences of our result.