On \(xD\)-generalizations of Stirling numbers and Lah numbers via graphs and rooks

Sen-Peng Eu¹ ; Tung-Shan Fu² ; Yu-Chang Liang² ; Tsai-Lien Wong³

¹Department of Mathematics, National Taiwan Normal University
²Department of Applied Mathematics, National Pingtung University
³Department of Applied Mathematics, National Sun Yat-sen University

The electronic journal of combinatorics, Tome 24 (2017) no. 2

Résumé

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 $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. 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 $\omega$, involving decreasing forest decompositions of quasi-threshold 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.

Détail

DOI : 10.37236/6699

Classification : 11B73, 05A10, 05A30, 05C31, 05E10
Mots-clés : Stirling numbers, Lah numbers, normal ordering problem, quasi-threshold graphs, rook numbers