Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid Gn of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QADn−1 of order n− 1 with the path P (2) n on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular that the characteristic polynomials of the above graphs satisfy the equality P(Gn)= P(P (2) n )[P(QADn−1)]2. On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described “linear squarishness” property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases. Research supported in part by NSF grant DMS-0500616. M. Ciucu ( ) Department of Mathematics, Indiana University, Bloomington, IN 47405, USA e-mail: [email protected] 494 J Algebr Comb (2008) 27: 493–538