Derangement Polynomials and Excedances of Type B

TL;DR

This paper extends the concept of derangement polynomials to type B permutations, establishing their properties and connections to Eulerian polynomials, including generating functions, asymptotic behavior, and symmetry features.

Contribution

It introduces a type B analogue of q-derangement polynomials, expanding their theoretical framework and analyzing their fundamental properties and distributions.

Findings

Derived the generating function formula for type B q-derangement polynomials

Established the Sturm sequence property and asymptotic normality

Showed the coefficients exhibit the spiral symmetry property

Abstract

Adopting the definition of excedances of type B due to Brenti, we give a type B analogue of the q-derangement polynomials. The connection between q-derangement polynomials and Eulerian polynomials naturally extends to the type B case. Based on this relation, we derive some basic properties of the q-derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the q-derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.