### Abstract

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra W = 〈x, D|Dx-xD = 1〉. Any word ω ∈ W with m x’s and n D’s can be expressed in the normally ordered form ω = x^{m−n}Σ_{k}
_{≥0}{ω/k} x kDk , where {ω/k} is known as the Stirling number of the second kind for the word ω. This study considers the expansions of restricted words ω in W over the sequences {(xD)^{k}}_{k}
_{≥0}and {xD^{k}x^{k−1}}_{k}
_{≥0}. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words ω, involving decreasing forest decompositions of quasithreshold graphs and non-attacking rook placements on Ferrers boards. Extended to q-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the q-deformed Weyl algebra.

Original language | English |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 2 |

Publication status | Published - 2017 Apr 13 |

### Fingerprint

### Keywords

- Lah numbers
- Normal ordering problem
- Quasithreshold graphs
- Rook numbers
- Stirling numbers

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*24*(2).

**On xD-generalizations of stirling numbers and lah numbers via graphs and rooks.** / Eu, Sen-Peng; Liang, Yu Chang; Fu, Tung Shan; Wong, Tsai Lien.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 24, no. 2.

}

TY - JOUR

T1 - On xD-generalizations of stirling numbers and lah numbers via graphs and rooks

AU - Eu, Sen-Peng

AU - Liang, Yu Chang

AU - Fu, Tung Shan

AU - Wong, Tsai Lien

PY - 2017/4/13

Y1 - 2017/4/13

N2 - This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra W = 〈x, D|Dx-xD = 1〉. Any word ω ∈ W with m x’s and n D’s can be expressed in the normally ordered form ω = xm−nΣk ≥0{ω/k} x kDk , where {ω/k} is known as the Stirling number of the second kind for the word ω. This study considers the expansions of restricted words ω in W over the sequences {(xD)k}k ≥0and {xDkxk−1}k ≥0. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words ω, involving decreasing forest decompositions of quasithreshold graphs and non-attacking rook placements on Ferrers boards. Extended to q-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the q-deformed Weyl algebra.

AB - This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra W = 〈x, D|Dx-xD = 1〉. Any word ω ∈ W with m x’s and n D’s can be expressed in the normally ordered form ω = xm−nΣk ≥0{ω/k} x kDk , where {ω/k} is known as the Stirling number of the second kind for the word ω. This study considers the expansions of restricted words ω in W over the sequences {(xD)k}k ≥0and {xDkxk−1}k ≥0. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words ω, involving decreasing forest decompositions of quasithreshold graphs and non-attacking rook placements on Ferrers boards. Extended to q-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the q-deformed Weyl algebra.

KW - Lah numbers

KW - Normal ordering problem

KW - Quasithreshold graphs

KW - Rook numbers

KW - Stirling numbers

UR - http://www.scopus.com/inward/record.url?scp=85018528768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018528768&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85018528768

VL - 24

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

ER -