17,074 research outputs found
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
On Enumeration of Conjugacy Classes of Coxeter Elements
In this paper we study the equivalence relation on the set of acyclic
orientations of a graph Y that arises through source-to-sink conversions. This
source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a
Coxeter group. We give a direct proof of a recursion for the number of
equivalence classes of this relation for an arbitrary graph Y using edge
deletion and edge contraction of non-bridge edges. We conclude by showing how
this result may also be obtained through an evaluation of the Tutte polynomial
as T(Y,1,0), and we provide bijections to two other classes of acyclic
orientations that are known to be counted in the same way. A transversal of the
set of equivalence classes is given.Comment: Added a few results about connections to the Tutte polynomia
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
{\Gamma}-species, quotients, and graph enumeration
The theory of {\Gamma}-species is developed to allow species-theoretic study
of quotient structures in a categorically rigorous fashion. This new approach
is then applied to two graph-enumeration problems which were previously
unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio
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