On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

TL;DR

This paper explores generalized Stirling and Lah numbers through combinatorial structures linked to words in the Weyl algebra, extending classical concepts with graph and rook placements, including q-analogues.

Contribution

It introduces new generalized numbers associated with words in the Weyl algebra and connects them to combinatorial structures like forests and rook placements, extending classical number concepts.

Findings

Coefficients correspond to generalized Stirling and Lah numbers.

Enumeration involves forest decompositions and rook placements.

q-analogues provide weighted combinatorial interpretations.

Abstract

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal order 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.