### Abstract

In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S^{1}(α,β,r) and S^{2}(α; β; r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α and r are nonnegative integers both of our (p; q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our (p; q)-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our (p; q)-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

Original language | English (US) |
---|---|

Pages (from-to) | 1-48 |

Number of pages | 48 |

Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 1 R |

State | Published - Nov 22 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Electronic Journal of Combinatorics*,

*11*(1 R), 1-48.

**Rook theory, generalized stirling numbers and (p, q)-analogues.** / Remmel, J. B.; Galloway, Michelle L.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 11, no. 1 R, pp. 1-48.

}

TY - JOUR

T1 - Rook theory, generalized stirling numbers and (p, q)-analogues

AU - Remmel, J. B.

AU - Galloway, Michelle L

PY - 2004/11/22

Y1 - 2004/11/22

N2 - In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S1(α,β,r) and S2(α; β; r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α and r are nonnegative integers both of our (p; q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our (p; q)-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our (p; q)-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

AB - In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S1(α,β,r) and S2(α; β; r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α and r are nonnegative integers both of our (p; q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our (p; q)-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our (p; q)-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

UR - http://www.scopus.com/inward/record.url?scp=8844281599&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=8844281599&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:8844281599

VL - 11

SP - 1

EP - 48

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1 R

ER -