193 research outputs found
New results on torus cube packings and tilings
We consider sequential random packing of integral translate of cubes
into the torus . Two special cases are of special
interest:
(i) The case which corresponds to a discrete case of tilings
(considered in \cite{cubetiling,book})
(ii) The case corresponds to a case of continuous tilings
(considered in \cite{combincubepack,book})
Both cases correspond to some special combinatorial structure and we describe
here new developments.Comment: 5 pages, conference pape
The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions
We prove that the number of cyclically symmetric, self-complementary plane
partitions contained in a cube of side equals the square of the number of
totally symmetric, self-complementary plane partitions contained in the same
cube, without explicitly evaluating either of these numbers. This appears to be
the first direct proof of this fact. The problem of finding such a proof was
suggested by Stanley
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
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
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
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