6 research outputs found
Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix
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 receive a parity: we construct a perfect or near-perfect
matching of tilings of opposite parities. Let be the black-to-white
adjacency matrix: we factor , where and are
lower and upper triangular matrices, is obtained from a larger
identity matrix by removing rows and columns and all entries of ,
and are equal to 0, 1 or -1.Comment: 20 pages, 17 figure
Abelian Sandpile Model on Symmetric Graphs
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
Enumeration of Matchings: Problems and Progress
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
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
Domino tiling, gene recognition and mice
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 186-192).by Lior Samuel Pachter.Ph.D