Plane partitions I: a generalization of MacMahon's formula

TL;DR

This paper generalizes MacMahon's formula for plane partitions by providing new product formulas for enumerating lozenge tilings of modified hexagonal regions with specific side-lengths and removed triangles.

Contribution

It introduces a novel generalization of MacMahon's formula for plane partitions, extending enumeration results to more complex hexagonal regions with triangular removals.

Findings

New product formulas for lozenge tilings of generalized hexagons

Extension of MacMahon's enumeration to regions with triangular cuts

Simplified bijections linking plane partitions and tilings

Abstract

The number of plane partitions contained in a given box was shown by MacMahon to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of side-lengths $a,b,c,a,b,c$ (in cyclic order) and angles of 120 degrees. We present a generalization in the case $b=c$ by giving simple product formulas enumerating lozenge tilings of regions obtained from a hexagon of side-lengths $a,b+k,b,a+k,b,b+k$ (where $k$ is an arbitrary non-negative integer) and angles of 120 degrees by removing certain triangular regions along its symmetry axis.