paper

The 1/k-Eulerian polynomials of type B

arXiv:2001.07833

Abstract

In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of polynomials, including combinatorial interpretations, recurrence relations and gamma-positivity are studied. In particular, we show that the 1/k-Eulerian polynomials of type B are gamma-positive when $k > 0$ . Moreover, we obtain the corresponding results for derangements of type B. We show that a type B 1/k-derangement polynomials $d_{n}^{B} (x; k)$ are bi-gamma-positive when $k \geq 1/2$ . In particular, we get a symmetric decomposition of $d_{n}^{B} (x; 1/2)$ in terms of the classical derangement polynomials.