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 $d_n^B\left(q\right)={\sum }_{\sigma \in D_n^B}{q}^{\mathrm{fmaj}\left(\sigma \right)}$ and $d_n^D\left(q\right)={\sum }_{\sigma \in D_n^D}{q}^{\mathrm{maj}\left(\sigma \right)}$ 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 $d_n=n!{\sum }_{k=0}^n{(-1)}^k/k!$ 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 $d_n^B\left(q\right)$ and $d_n^D\left(q\right)$ as determinants of order $n+1$.