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

Tiles and colors

Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles are rectangles and isosceles triangles. Some of these models have been solved recently by means of Bethe Ansatz. We discuss the question why only these models in a larger family are solvable, and we search for the Yang-Baxter structure behind their integrablity. In this quest we find the Bethe Ansatz solution of the problem of coloring the edges of the square lattice in four colors, such that edges of the same color never meet in the same vertex.Comment: 18 pages, 3 figures (in 5 eps files

Tetratic Order in the Phase Behavior of a Hard-Rectangle System

Previous Monte Carlo investigations by Wojciechowski \emph{et al.} have found two unusual phases in two-dimensional systems of anisotropic hard particles: a tetratic phase of four-fold symmetry for hard squares [Comp. Methods in Science and Tech., 10: 235-255, 2004], and a nonperiodic degenerate solid phase for hard-disk dimers [Phys. Rev. Lett., 66: 3168-3171, 1991]. In this work, we study a system of hard rectangles of aspect ratio two, i.e., hard-square dimers (or dominos), and demonstrate that it exhibits a solid phase with both of these unusual properties. The solid shows tetratic, but not nematic, order, and it is nonperiodic having the structure of a random tiling of the square lattice with dominos. We obtain similar results with both a classical Monte Carlo method using true rectangles and a novel molecular dynamics algorithm employing rectangles with rounded corners. It is remarkable that such simple convex two-dimensional shapes can produce such rich phase behavior. Although we have not performed exact free-energy calculations, we expect that the random domino tiling is thermodynamically stabilized by its degeneracy entropy, well-known to be $1.79k_{B}$ per particle from previous studies of the dimer problem on the square lattice. Our observations are consistent with a KTHNY two-stage phase transition scenario with two continuous phase transitions, the first from isotropic to tetratic liquid, and the second from tetratic liquid to solid.Comment: Submitted for publicatio

Enumeration of tilings of quartered Aztec rectangles

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindstr\"{o}m-Gessel-Viennot methodology to find the number of tilings of a quartered lozenge hexagon.Comment: 28 page

Applications of Graphical Condensation for Enumerating Matchings and Tilings

A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text revised, but content essentially the sam