Perfect Matchings of Trimmed Aztec Rectangles
Keywords: perfect matching, tiling, dual graph, Aztec rectangle, graphical condensation, hexagonal dungeon
AbstractWe consider several new families of subgraphs of the square grid whose matchings are enumerated by powers of several small prime numbers: $2$, $3$, $5$, and $11$. Our graphs are obtained by trimming two opposite corners of an Aztec rectangle. The result yields a proof of a conjecture posed by Ciucu. In addition, we reveal a hidden connection between our graphs and the hexagonal dungeons introduced by Blum.