Some determinantal representations of derangement numbers and polynomials.

Munarini (J Integer Seq 23: Article 20.3.8, 2020) recently showed that the derangement polynomial d n (q) = ∑ σ ∈ D n q maj (σ) is expressible as the determinant of either an n × n tridiagonal matrix or an n × n lower Hessenberg matrix. Qi et al. (Cogent Math 3:1232878, 2016) showed that the classical derangement number d n = n ! ∑ k = 0 n (- 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) = ∑ σ ∈ D n B q fmaj (σ) studied previously by Chow (Sém Lothar Combin 55:B55b, 2006), and the type D derangement polynomial d n D (q) = ∑ σ ∈ D n D q maj (σ) 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. [ABSTRACT FROM AUTHOR]