On the numbers of perfect matchings of trimmed Aztec rectangles

TL;DR

This paper investigates the enumeration of perfect matchings in new graph families derived from Aztec rectangles, revealing they are counted by powers of specific primes and establishing connections to hexagonal dungeons.

Contribution

It introduces new graph families from Aztec rectangles, proves their perfect matchings are counted by prime powers, and uncovers a relation to hexagonal dungeon graphs.

Findings

Perfect matchings are counted by powers of 2, 3, 5, and 11.

Established a proof for a conjecture by Ciucu.

Discovered a relation to hexagonal dungeon graphs.

Abstract

We consider several new families of graphs obtained from Aztec rectangle and augmented Aztec rectangle graphs by trimming two opposite corners. We prove that the perfect matchings of these new graphs are enumerated by powers of $2$, $3$, $5$, and $11$. The result yields a proof of a conjectured posed by Ciucu. In addition, we reveal a hidden relation between our graphs and the hexagonal dungeons introduced by Blum.