A $q$-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

TL;DR

This paper generalizes MacMahon's classical theorem by providing a $q$-enumeration formula for lozenge tilings of hexagons with four adjacent boundary triangles removed, extending combinatorial enumeration techniques.

Contribution

It introduces a new family of hexagons with boundary modifications and derives a $q$-enumeration formula for their lozenge tilings, expanding classical tiling enumeration results.

Findings

Derived a $q$-enumeration formula for the new hexagon family.

Extended MacMahon's theorem to boundary-modified hexagons.

Provided combinatorial insights into tilings with boundary removals.

Abstract

MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a $q$-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we generalize MacMahon's classical theorem by $q$-enumerating lozenge tilings of a new family of hexagons with four adjacent triangles removed from their boundary.