The Eulerian Distribution on k-Colored Involutions

Abstract

Let ${\mathcal {I}}_{n,k}$ be the set of k-colored involutions of order n and ${\mathcal {J}}_{n,k}$ be the set of k-colored involutions in ${\mathcal {I}}_{n,k}$ without fixed points. Denote by $des(\pi ,c)$ the number of descents of k-colored permutations $(\pi ,c)$. In this paper, it is proved that the following polynomials

$$\begin{aligned} I_{n,k}(x)= & {} \sum \limits _{(\pi ,c)\in {\mathcal {I}}_{n,k}}x^{des(\pi ,c)} ( n\ge 1, k\ge 1)\text { and }\\ J_{n,k}(x)= & {} \sum \limits _{(\pi ,c)\in {\mathcal {J}}_{n,k}}x^{des(\pi ,c)} ( n\ge 1, k\ge 2 ) \end{aligned}$$

are $\gamma$-positive.