Some determinantal representations of derangement numbers and polynomials

Abstract

Munarini (J Integer Seq 23: Article 20.3.8, 2020) recently showed that the derangement polynomial $d_n(q)=\sum _{\sigma \in {\mathcal {D}}_n}q^{{{\,\textrm{maj}\,}}(\sigma )}$ is expressible as the determinant of either an $n\times n$ tridiagonal matrix or an $n\times n$ lower Hessenberg matrix. Qi et al. (Cogent Math 3:1232878, 2016) showed that the classical derangement number $d_n=n!\sum _{k=0}^n\frac{(-1)^k}{k!}$ is expressible as a tridiagonal determinant of order $n+1$. We show in this work that similar determinantal expressions exist for the type B derangement polynomial $d_n^B(q)=\sum _{\sigma \in {\mathcal {D}}_n^B}q^{{{\,\textrm{fmaj}\,}}(\sigma )}$ studied previously by Chow (Sém Lothar Combin 55:B55b, 2006), and the type D derangement polynomial $d_n^D(q)=\sum _{\sigma \in {\mathcal {D}}_n^D}q^{{{\,\textrm{maj}\,}}(\sigma )}$ studied recently by Chow (Taiwanese J Math 27(4):629–646, 2023). Representations of the types B and D derangement numbers $d_n^B$ and $d_n^D$ as determinants of order $n+1$ are also presented.