Some identities involving $q$-Stirling numbers of the second kind in type B

TL;DR

This paper explores new identities involving $q$-Stirling numbers of the second kind in type B, providing combinatorial and analytical proofs, and extends results to types A and D.

Contribution

It introduces a type B analogue of a classical identity linking $q$-Stirling numbers and Eulerian numbers, with proofs and extensions to other types.

Findings

Established a $q$-analogue of a classical identity in type B.

Provided combinatorial and analytical proofs for the identities.

Derived new $q$-identities for types A, B, and D.

Abstract

The recent interest in $q$-Stirling numbers of the second kind in type B prompted us to give a type B analogue of a classical identity connecting the $q$-Stirling numbers of the second kind and Carlitz's major $q$-Eulerian numbers, which turns out to be a $q$-analogue of an identity due to Bagno, Biagioli and Garber. We provide a combinatorial proof of this identity and an analytical proof of a more general identity for colored permutations. In addition, we prove some $q$-identities about the $q$-Stirling numbers of the second kind in types A, B and D.