Lozenge tilings of hexagons with arbitrary dents

TL;DR

This paper generalizes the enumeration of lozenge tilings of hexagons with boundary dents, providing formulas for cases with arbitrary boundary removals, extending previous specific and partial results.

Contribution

It introduces a comprehensive formula for lozenge tilings of hexagons with arbitrary boundary dents, broadening the scope of previous specific cases.

Findings

Derived a general enumeration formula for boundary-dented hexagons.

Extended previous results to arbitrary boundary removals.

Provided combinatorial insights into lozenge tilings with complex boundary conditions.

Abstract

Eisenk"olbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.