Pseudo-centrosymmetric matrices, with applications to counting perfect matchings
Abstract
We consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we define A to be pseudo-centrosymmetric with respect to K; we show that the determinant of every even-order pseudo-centrosymmetric matrix is the sum of two squares over F, as long as -1 is not a square in F. When a pseudo-centrosymmetric matrix A contains only integral entries and is pseudo-centrosymmetric with respect to a matrix with rational entries, the determinant of A is the sum of two integral squares. This result, when specialized to when K is the even-order alternating exchange matrix, applies to enumerative combinatorics. Using solely matrix-based methods, we reprove a weak form of Jockusch's theorem for enumerating perfect matchings of 2-even symmetric graphs. As a corollary, we reprove that the number of domino tilings of regions known as Aztec diamonds and Aztec pillows is a sum of two integral squares.
Keywords
Cite
@article{arxiv.math/0609622,
title = {Pseudo-centrosymmetric matrices, with applications to counting perfect matchings},
author = {Christopher R. H. Hanusa},
journal= {arXiv preprint arXiv:math/0609622},
year = {2007}
}
Comments
v1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures. Reworked so that linear algebraic results are over a field not of characteristic 2, not over the real numbers. Accepted, Linear Algebra and its Applications