On the structure of the lattice of noncrossing partitions
Discrete Mathematics
Enumeration of functions from posets to chains
European Journal of Combinatorics
Combinatorial statistics on non-crossing partitions
Journal of Combinatorial Theory Series A
Octobasic Laguerre polynomials and permutation statistics
Journal of Computational and Applied Mathematics
On the Neggers-Stanley conjecture and the Eulerian polynomials
Journal of Combinatorial Theory Series A
Permutation trees and variation statistics
European Journal of Combinatorics
Arbres minimax et polynômes d'André
Advances in Applied Mathematics
Symmetry and unimodality in t-stack sortable permutations
Journal of Combinatorial Theory Series A
On Operators on Polynomials Preserving Real-Rootedness and the Neggers-Stanley Conjecture
Journal of Algebraic Combinatorics: An International Journal
On the Charney-Davis and Neggers-Stanley conjectures
Journal of Combinatorial Theory Series A
Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres
Discrete & Computational Geometry
The Eulerian distribution on involutions is indeed unimodal
Journal of Combinatorial Theory Series A
Permutation tableaux and permutation patterns
Journal of Combinatorial Theory Series A
Permutation statistics on involutions
European Journal of Combinatorics
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
European Journal of Combinatorics
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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 non-negative expansion in the basis {t^i(1+t)^n^-^1^-^2^i}"i"="0^m, m=@?(n-1)/2@?. This property implies symmetry and unimodality. We prove that the action is invariant under stack sorting which strengthens recent unimodality results of Bona. 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, Steingrimsson 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,@w)-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. On restricting to the set of stack sortable permutations we recover a result of Kreweras.