Some Identities Involving $q$-Stirling Numbers of the Second Kind in Type B

Abstract

The recent interest in type B $q$-Stirling numbers of the second kind 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 algebraic 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.