Symmetric unimodal expansions of excedances in colored permutations

Abstract

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.

27 pages, 10 figures

References in corpus (4)

• Statistics on Wreath Products
• On the excedance number of colored permutation groups
• A symmetric unimodal decomposition of the derangement polynomial of type $B$
• Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\ZZ_r \wr \mathfrak{S}_n$

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• Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions
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• Alternating Eulerian polynomials and left peak polynomials
• On two unimodal descent polynomials
• Excedance-type polynomials and gamma-positivity
• A Hamilton Cycle in the $k$-Sided Pancake Network
• Eulerian pairs and Eulerian recurrence systems
• Brändén's $(p,q)$-Eulerian polynomials, André permutations and continued fractions
• Eulerian polynomials and excedance statistics
• A new encoding of permutations by Laguerre histories
• Eulerian polynomials, Stirling permutations and increasing trees
• Signed Countings of types B and D permutations and $t,q$-Euler Numbers
• The gamma-positivity of Eulerian polynomials and succession statistics
• The 1/k-Eulerian polynomials of type B