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Combinatorics of q,r-analogues of Stirling numbers of type B

Eli Bagno, David Garber

Abstract

Stirling number of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters which realize $q$-analogues and Broder's $r$-variants of Stirling numbers of type $B$ of both kinds, which count signed set partitions and signed permutations respectively. Applications to orthogonality relations and power sums are given.

Combinatorics of q,r-analogues of Stirling numbers of type B

Abstract

Stirling number of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters which realize -analogues and Broder's -variants of Stirling numbers of type of both kinds, which count signed set partitions and signed permutations respectively. Applications to orthogonality relations and power sums are given.
Paper Structure (23 sections, 19 theorems, 120 equations)

This paper contains 23 sections, 19 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Restricted growth words for Stirling number of type A of the second kind
  4. Set partitions of type B
  5. The group of signed permutations
  6. A q,r-analogue of the Stirling number of the second kind
  7. The r-variant
  8. Restricted growth words for set partitions of type B and their weights
  9. The q,r-variant
  10. Restricted growth words and q,r-Stirling numbers of the first kind
  11. Type A
  12. Restricted growth words for permutations
  13. A q-analogue
  14. A q,r-analogue
  15. Type B
  16. ...and 8 more sections

Key Result

Proposition 2.4

For each $1 \leq k < n$,

Theorems & Definitions (75)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 65 more