Some determinantal representations of derangement polynomials of types B and D

Abstract

Chow (2024) recently computed expressions of the types B and D derangement polynomials dnB(q)=∑σ∈DnBqfmaj(σ) and dnD(q)=∑σ∈DnDqmaj(σ) as tridiagonal and lower Hessenberg determinants of order n. Qi, Wang, and Guo (2016), based on a determinantal formula for the nth derivative of a quotient of two functions, derived an expression of the classical derangement number dn=n!∑k=0n(−1)k/k! as a tridiagonal determinant of order n+1. By q-extending the approach of Qi et al., we present in this work yet another determinantal expressions of dnB(q) and dnD(q) as determinants of order n+1.