Séminaire Lotharingien de Combinatoire, 78B.26 (2017), 12 pp.

Tomack Gilmore

Holey Matrimony: Marrying Two Approaches to a Tiling Problem

Abstract. Consider an hexagonal region on the triangular lattice, the interior of which contains a number of holes. This extended abstract outlines a recent result by the author that marries together two separate approaches to counting tilings in order to express the number of rhombus tilings of a holey hexagon (subject to very mild restrictions) as a determinant whose size is dependent only on the regions that have been removed. The main result follows from explicitly deriving the (i,j)-entries of the inverse Kasteleyn matrix corresponding to certain sub-graphs of the hexagonal lattice. This generalises a number of known results and may well lead to a proof of Ciucu's electrostatic conjecture for the most general family of holes to date.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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