56 research outputs found

    Lambda-determinants and domino-tilings

    Get PDF
    Consider the 2n2n-by-2n2n matrix M=(mi,j)i,j=12nM=(m_{i,j})_{i,j=1}^{2n} with mi,j=1m_{i,j} = 1 for i,ji,j satisfying ∣2i−2n−1∣+∣2j−2n−1∣≤2n|2i-2n-1|+|2j-2n-1| \leq 2n and mi,j=0m_{i,j} = 0 for all other i,ji,j, consisting of a central diamond of 1's surrounded by 0's. When n≥4n \geq 4, the λ\lambda-determinant of the matrix MM (as introduced by Robbins and Rumsey) is not well-defined. However, if we replace the 0's by tt's, we get a matrix whose λ\lambda-determinant is well-defined and is a polynomial in λ\lambda and tt. The limit of this polynomial as t→0t \to 0 is a polynomial in λ\lambda whose value at λ=1\lambda=1 is the number of domino tilings of a 2n2n-by-2n2n square.Comment: 4 pages; to appear in a special issue of Advances in Applied Mathematics honoring David P. Robbin

    A Generalization of Aztec Diamond Theorem, Part II

    Full text link
    The author gave a proof of a generalization of the Aztec diamond theorem for a family of 44-vertex regions on the square lattice with southwest-to-northeast diagonals drawn in (Electron. J. Combin., 2014) by using a bijection between tilings and non-intersecting lattice paths. In this paper, we use Kuo graphical condensation to give a new proof.Comment: 11 pages and 7 figure

    Graphical condensation of plane graphs: a combinatorial approach

    Get PDF
    The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc

    A generalization of Aztec diamond theorem, part I

    Get PDF
    We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic Combinatoric, 1992) by showing that the numbers of tilings of a certain family of regions in the square lattice with southwest-to-northeast diagonals drawn in are given by powers of 2. We present a proof for the generalization by using a bijection between domino tilings and non-intersecting lattice paths.Comment: 18 page
    • …
    corecore