28 research outputs found
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
The Tchebyshev transforms of the first and second kind
We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction