Singular polynomials of generalized Kasteleyn matrices

Details Singular polynomials of generalized Kasteleyn matrices Singular polynomials of generalized Kasteleyn matrices

2000-12-30

arxiv.org/abs/math/0101002v1

Journal of Algebraic Combinatorics, 16(2): 195-207; Sep 2002

Mathematics

Combinatorics

15 pages, 4 figures

Scientific paper

Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK^\ast (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.

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