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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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