Excedance-type polynomials and gamma-positivity

TL;DR

This paper systematically studies excedance-type polynomials, proving gamma-positivity and unimodality properties, and explores their combinatorial interpretations and relationships with various permutation statistics.

Contribution

It provides new conditions for polynomial properties, proves bi-gamma-positivity of cyc q-Eulerian polynomials, and unifies various recent results in permutation statistics.

Findings

Cyc q-Eulerian polynomials are bi-gamma-positive.

Fix and cyc (p,q)-Eulerian polynomials are alternatingly increasing.

Established relationships between (p,q)-Eulerian and multivariate Eulerian polynomials.

Abstract

The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let $p\in [0,1]$ and $q\in [0,1]$ be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, and so they are unimodal with modes in the middle, where fix and cyc are the fixed point and cycle statistics. When p=1 and q=1/2, we find a combinatorial interpretation of the bi-gamma-coefficients of the (p,q)-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. In particular, we establish the relationships between the (p,q)-Eulerian polynomials and some multivariate Eulerian polynomials. Our results unify and generalize a variety of recent results.