A dual of MacMahon's theorem on plane partitions

TL;DR

This paper presents a new product formula for counting lozenge tilings of the exterior of a concave hexagon, providing a dual perspective to MacMahon's classical theorem on plane partitions.

Contribution

It introduces a novel combinatorial formula for tilings of a concave hexagon's exterior, extending MacMahon's theorem to a new geometric configuration.

Findings

Derived a product formula for the exterior of a concave hexagon

Extended MacMahon's theorem to a dual geometric setting

Provided combinatorial enumeration for new tiling configurations

Abstract

A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the {\it exterior} of a concave hexagon obtained by turning 120 degrees after drawing each side (MacMahon's hexagon is obtained by turning 60 degrees after each step).