Abstract
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.
Highlights
Introduction and main resultsA perfect matching of a graph G is a collection of edges of G so that each vertex is covered by precisely one edge in the collection
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
The field of enumeration of perfect matchings date back to the early 1960’s when Kasteleyn [8] and Temperley and Fisher [29] found the number of tilings of a rectangle on the square grid
Summary
Introduction and main resultsA perfect matching of a graph G is a collection of edges of G so that each vertex is covered by precisely one edge in the collection. We use the Kuo’s Condensation Theorem to prove that the numbers of perfect matchings of the six families of graphs A(ai,)b,c’s and Fa(,ib),c’s, for i = 1, 2, 3, satisfy the following six recurrences (with some constraints).
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