125 research outputs found
The short toric polynomial
We introduce the short toric polynomial associated to a graded Eulerian
poset. This polynomial contains the same information as the two toric
polynomials introduced by Stanley, but allows different algebraic
manipulations. The intertwined recurrence defining Stanley's toric polynomials
may be replaced by a single recurrence, in which the degree of the discarded
terms is independent of the rank. A short toric variant of the formula by Bayer
and Ehrenborg, expressing the toric -vector in terms of the -index, may
be stated in a rank-independent form, and it may be shown using weighted
lattice path enumeration and the reflection principle. We use our techniques to
derive a formula expressing the toric -vector of a dual simplicial Eulerian
poset in terms of its -vector. This formula implies Gessel's formula for the
toric -vector of a cube, and may be used to prove that the nonnegativity of
the toric -vector of a simple polytope is a consequence of the Generalized
Lower Bound Theorem holding for simplicial polytopes.Comment: Minor correction
Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function
A partially ordered set is r-thick if every nonempty open interval contains
at least r elements. This paper studies the flag vectors of graded, r-thick
posets and shows the smallest convex cone containing them is isomorphic to the
cone of flag vectors of all graded posets. It also defines a k-analogue of the
Mobius function and k-Eulerian posets, which are 2k-thick. Several
characterizations of k-Eulerian posets are given. The generalized
Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A
new inequality is proved to be valid and sharp for rank 8 Eulerian posets
The toric h-vector of a cubical complex in terms of noncrossing partition statistics
This paper introduces a new and simple statistic on noncrossing partitions
that expresses each coordinate of the toric -vector of a cubical complex,
written in the basis of the Adin -vector entries, as the total weight of all
noncrossing partitions. The same model may also be used to obtain a very simple
combinatorial interpretation of the contribution of a cubical shelling
component to the toric -vector. In this model, a strengthening of the
symmetry expressed by the Dehn-Sommerville equations may be derived from the
self-duality of the noncrossing partition lattice, exhibited by the involution
of Simion and Ullman
Relative Tutte polynomials of tensor products of colored graphs
The tensor product of a graph and a pointed graph
(containing one distinguished edge) is obtained by identifying each edge of
with the distinguished edge of a separate copy of , and then
removing the identified edges. A formula to compute the Tutte polynomial of a
tensor product of graphs was originally given by Brylawski. This formula was
recently generalized to colored graphs and the generalized Tutte polynomial
introduced by Bollob\'as and Riordan. In this paper we generalize the colored
tensor product formula to relative Tutte polynomials of relative graphs,
containing zero edges to which the usual deletion-contraction rules do not
apply. As we have shown in a recent paper, relative Tutte polynomials may be
used to compute the Jones polynomial of a virtual knot
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