# Some identities involving second kind Stirling numbers of types $B$ and $D$

## TL;DR

This paper generalizes classical identities involving second kind Stirling numbers of types B and D, connecting them with Eulerian numbers and polynomial bases, and extends these results to colored permutation groups.

## Contribution

It introduces new identities for Stirling numbers of types B and D and extends these identities to the group of colored permutations G_{m,n}.

## Key findings

- Generalized identities linking Stirling and Eulerian numbers.
- Interpreted Stirling numbers as transition matrix entries.
- Extended identities to colored permutation groups.

## Abstract

Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.