Selberg integrals, Askey-Wilson polynomials and lozenge tilings of a hexagon with a triangular hole

TL;DR

This paper derives an explicit formula for weighted lozenge tilings of a hexagon with a triangular hole, generalizing previous conjectures by connecting combinatorics with Askey-Wilson polynomials and Selberg integrals.

Contribution

It introduces a new explicit formula for weighted tilings with an arbitrary triangular hole, extending prior results and employing Askey-Wilson polynomials to link discrete and continuous integrals.

Findings

Derived explicit weighted enumeration formula for lozenge tilings

Generalized previous conjectures to arbitrary triangle positions

Connected combinatorial tilings with Askey-Wilson polynomials and Selberg integrals

Abstract

We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This proves and generalizes conjectures of Ciucu et al., who considered the case of plain enumeration when the triangle is located at or very near the center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete and continuous Selberg-type integrals.