35 research outputs found
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
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
Divisors and specializations of Lucas polynomials
Three-term recurrences have infused stupendous amount of research in a broad
spectrum of the sciences, such as orthogonal polynomials (in special functions)
and lattice paths (in enumerative combinatorics). Among these are the Lucas
polynomials, which have seen a recent true revival. In this paper one of the
themes of investigation is the specialization to the Pell and Delannoy numbers.
The underpinning motivation comprises primarily of divisibility and symmetry.
One of the most remarkable findings is a structural decomposition of the Lucas
polynomials into what we term as flat and sharp analogs.Comment: Minor typos are fixed, new references are added. To appear in Journal
of Combinatoric
A counterexample to the periodic tiling conjecture (announcement)
The periodic tiling conjecture asserts that any finite subset of a lattice
which tiles that lattice by translations, in fact tiles
periodically. We announce here a disproof of this conjecture for sufficiently
large , which also implies a disproof of the corresponding conjecture for
Euclidean spaces . In fact, we also obtain a counterexample in a
group of the form for some finite abelian . Our
methods rely on encoding a certain class of "-adically structured functions"
in terms of certain functional equations
-regularity of the -adic valuation of the Fibonacci sequence
We show that the -adic valuation of the sequence of Fibonacci numbers is a
-regular sequence for every prime . For , we determine that
the rank of this sequence is , where is the
restricted period length of the Fibonacci sequence modulo .Comment: 7 pages; publication versio
A counterexample to the periodic tiling conjecture
The periodic tiling conjecture asserts that any finite subset of a lattice
which tiles that lattice by translations, in fact tiles
periodically. In this work we disprove this conjecture for sufficiently large
, which also implies a disproof of the corresponding conjecture for
Euclidean spaces . In fact, we also obtain a counterexample in a
group of the form for some finite abelian -group
. Our methods rely on encoding a "Sudoku puzzle" whose rows and other
non-horizontal lines are constrained to lie in a certain class of "-adically
structured functions", in terms of certain functional equations that can be
encoded in turn as a single tiling equation, and then demonstrating that
solutions to this Sudoku puzzle exist but are all non-periodic.Comment: 50 pages, 13 figures. Minor changes and additions of new reference