Singular Polynomials of Generalized Kasteleyn Matrices
Nicolau C. Saldanha
DOI: 10.1023/A:1021129230295
Abstract
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.
Pages: 195–207
Keywords: domino tilings; dimers; kasteleyn matrix; singular values
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References
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2. P.W. Kasteleyn, “The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice,” Phisica 27 (1961), 1209-1225.
3. E.H. Lieb and M. Loss, “Fluxes, Laplacians and Kasteleyn's theorem,” Duke Math. Jour. 71 (1993), 337-363.
4. J. Propp, “Enumeration of matchings, problems and progress,” in New Perspectives in Algebraic Combinatorics, Louis J. Billera, Anders Bjrner, Curtis Greene, Rodica Simion, and Richard P. Stanley (Eds.), MSRI Publications, Vol. 38, 1999.
5. N.C. Saldanha and C. Tomei, “An overview of domino and lozenge tilings,” Resenhas IME-USP 2(2) (1995), 239-252.