The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a $(p,q)$-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The $(p,q)$-analogue unifies and generalizes our recent results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).Comment: 19 pages, 2 figure

Symmetric unimodal expansions of excedances in colored permutations

We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $\gamma$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the literature.Comment: 27 pages, 10 figure

A survey of subdivisions and local hh-vectors

The enumerative theory of simplicial subdivisions (triangulations) of simplicial complexes was developed by Stanley in order to understand the effect of such subdivisions on the $h$-vector of a simplicial complex. A key role there is played by the concept of a local $h$-vector. This paper surveys some of the highlights of this theory and some recent developments, concerning subdivisions of flag homology spheres and their $\gamma$-vectors. Several interesting examples and open problems are discussed.Comment: 13 pages, 3 figures; minor changes and update

Actions on permutations and unimodality of descent polynomials

We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which strengthens recent unimodality results of B\'ona. We prove that the generalized permutation patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the $(P,\omega)$-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. When restricted to the set of stack-sortable permutations we recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi