Double Aztec Rectangles

TL;DR

This paper explores the relationship between lozenge and domino tilings by introducing a new family of regions formed from two Aztec rectangles, providing a product formula for their tilings and linking it to MacMahon's plane partition enumeration.

Contribution

It introduces a novel family of tiling regions combining Aztec rectangles and derives a simple product formula for their tilings, connecting to MacMahon's enumeration.

Findings

Derived a product formula for tilings of new regions

Established a connection to MacMahon's q-enumeration

Extended tiling enumeration techniques to combined Aztec regions

Abstract

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's $q$-enumeration of plane partitions fitting in a given box