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

Ming-Jian Ding, Jiang Zeng

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.

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

Abstract

The recent interest in -Stirling numbers of the second kind in type B prompted us to give a type B analogue of a classical identity connecting the -Stirling numbers of the second kind and Carlitz's major -Eulerian numbers, which turns out to be a -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 -identities about the -Stirling numbers of the second kind in types A, B and D.
Paper Structure (8 sections, 103 equations)

This paper contains 8 sections, 103 equations.

Table of Contents

  1. Introduction
  2. Combinatorial proof of Theorem \ref{['thm+stirling+frobenius+B']}
  3. Permutations of type B
  4. Ordered set partitions of type B
  5. $q$-Stirling numbers of the second kind in type D
  6. Two $q$-identities about the $q$-Stirling numbers of the second kind
  7. Falling factorials and $q$-Stirling numbers of the second kind in type D
  8. Generalization to colored permutations

Theorems & Definitions (19)

  • Remark 1.1
  • Remark 2.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['thm+stirling+frobenius+B']}
  • Definition 3.1
  • ...and 9 more