Enumeration of lozenge tilings of halved hexagons with a boundary defect

TL;DR

This paper extends enumeration formulas for lozenge tilings of hexagonal regions with boundary defects, using graphical condensation and factorization techniques, and generalizes MacMahon's plane partition formula.

Contribution

It introduces new enumeration formulas for lozenge tilings of halved hexagons with boundary defects, generalizing prior results and applying advanced combinatorial methods.

Findings

Derived a formula for lozenge tilings with boundary defects

Extended MacMahon's plane partition formula

Applied graphical condensation and factorization techniques

Abstract

We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed, using Kuo's graphical condensation method. Additionally, we prove a formula for a weighted version of the given region. The result also extends work of Ciucu and Fischer. By applying the factorization theorem of Ciucu, we are also able to generalize a special case of MacMahon's boxed plane partition formula.