7 research outputs found

    Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

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    We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk Δ\Delta receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let BΔB_\Delta be the black-to-white adjacency matrix: we factor BΔ=LD~UB_\Delta = L\tilde DU, where LL and UU are lower and upper triangular matrices, D~\tilde D is obtained from a larger identity matrix by removing rows and columns and all entries of LL, D~\tilde D and UU are equal to 0, 1 or -1.Comment: 20 pages, 17 figure

    Abelian Sandpile Model on Symmetric Graphs

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    The abelian sandpile model, or chip firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sandpile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid

    Domino tiling, gene recognition and mice

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    Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 186-192).by Lior Samuel Pachter.Ph.D

    Enumeration of Matchings: Problems and Progress

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    This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley), Mathematical Science Research Institute publication #37, Cambridge University Press, 199

    Combinatorial Approaches and Conjectures for 2-Divisibility Problems Concerning Domino Tilings of Polyominoes

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    We give the first complete combinatorial proof of the fact that the number of domino tilings of the 2n 2n square grid is of the form 2 n (2k + 1) 2 , thus settling a question raised in [4] . The proof lends itself naturally to some interesting generalizations, and leads to a number of new conjectures
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