2000/12/30 by Nicolau C. Saldanha
Mathematics · #math.CO #msc:05B45
published as Journal of Algebraic Combinatorics, 16(2): 195-207; Sep 2002 · 15 pages, 4 figures
arxiv created 2000/12/30 · arxiv updated 2009/11/30
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^∗ (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.