78 research outputs found
A factorization theorem for lozenge tilings of a hexagon with triangular holes
In this paper we present a combinatorial generalization of the fact that the
number of plane partitions that fit in a box is equal to
the number of such plane partitions that are symmetric, times the number of
such plane partitions for which the transpose is the same as the complement. We
use the equivalent phrasing of this identity in terms of symmetry classes of
lozenge tilings of a hexagon on the triangular lattice. Our generalization
consists of allowing the hexagon have certain symmetrically placed holes along
its horizontal symmetry axis. The special case when there are no holes can be
viewed as a new, simpler proof of the enumeration of symmetric plane
partitions.Comment: 20 page
Enumeration of lozenge tilings of hexagons with cut off corners
Motivated by the enumeration of a class of plane partitions studied by
Proctor and by considerations about symmetry classes of plane partitions, we
consider the problem of enumerating lozenge tilings of a hexagon with ``maximal
staircases'' removed from some of its vertices. The case of one vertex
corresponds to Proctor's problem. For two vertices there are several cases to
consider, and most of them lead to nice enumeration formulas. For three or more
vertices there do not seem to exist nice product formulas in general, but in
one special situation a lot of factorization occurs, and we pose the problem of
finding a formula for the number of tilings in this case.Comment: 23 pages, AmS-Te
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
The interaction of a gap with a free boundary in a two dimensional dimer system
Let be a fixed vertical lattice line of the unit triangular lattice in
the plane, and let \Cal H be the half plane to the left of . We
consider lozenge tilings of \Cal H that have a triangular gap of side-length
two and in which is a free boundary - i.e., tiles are allowed to
protrude out half-way across . We prove that the correlation function of
this gap near the free boundary has asymptotics ,
, where is the distance from the gap to the free boundary. This
parallels the electrostatic phenomenon by which the field of an electric charge
near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te
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