Perfect Matchings of Trimmed Aztec Rectangles

Tri Lai


We 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.


perfect matching; tiling; dual graph; Aztec rectangle; graphical condensation; hexagonal dungeon

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