125 research outputs found
On FPL configurations with four sets of nested arches
The problem of counting the number of Fully Packed Loop (FPL) configurations
with four sets of a,b,c,d nested arches is addressed. It is shown that it may
be expressed as the problem of enumeration of tilings of a domain of the
triangular lattice with a conic singularity. After reexpression in terms of
non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a
formula as a sum of determinants. This is made quite explicit when
min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates
the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function
adde
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
Exact solution of the dimer model: Corner free energy, correlation functions and combinatorics
In this work, some classical results of the pfaffian theory of the dimer
model based on the work of Kasteleyn, Fisher and Temperley are introduced in a
fermionic framework. Then we shall detail the bosonic formulation of the model
{\it via} the so-called height mapping and the nature of boundary conditions is
unravelled. The complete and detailed fermionic solution of the dimer model on
the square lattice with an arbitrary number of monomers is presented, and
finite size effect analysis is performed to study surface and corner effects,
leading to the extrapolation of the central charge of the model. The solution
allows for exact calculations of monomer and dimer correlation functions in the
discrete level and the scaling behavior can be inferred in order to find the
set of scaling dimensions and compare to the bosonic theory which predict
particular features concerning corner behaviors. Finally, some combinatorial
and numerical properties of partition functions with boundary monomers are
discussed, proved and checked with enumeration algorithms.Comment: Final version to be published in Nuclear Physics B (53 pages and a
lot of figures
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
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