Trees and Matchings

In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the ``square-octagon'' lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon, our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1

Signed Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations

Enumeration of Hybrid Domino-Lozenge Tilings

We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.Comment: 35 pages, 31 figure