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 $W=\langle x,D|Dx-xD=1\rangle$. 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.
Highlights
The Stirling numbers of both kinds and the Lah numbers are ubiquitous in combinatorics.In this paper, we study the generalizations of these numbers in association with the normal order problem in the Weyl algebra W generated by two operators x and D with the relationDx − xD = 1
We study the generalizations of these numbers in association with the normal order problem in the Weyl algebra W generated by two operators x and D with the relation
Considering the expansion of the specific word ω =n in Eq (10), we present a new q-Stirling number of the second kind (Theorem 3.5), which is different from the one introduced by Carlitz [4]
Summary
The Stirling numbers of both kinds and the Lah numbers are ubiquitous in combinatorics. For any word ω in W with m x’s and n D’s, a q-analogue of the xD-Stirling number of the second kind, denoted as ω k q, is defined by the following expansion ω = xm−n ω xkDk. Varvak [18] gave a combinatorial interpretation for ω k q by defining an inversion statistic for the rook placements on the Ferrers board associated with ω. Considering the expansion of the word ω = xnDn in Eq (10), we present a new q-Lah number, n k q, realized by weighted decreasing forest decompositions of a complete graph (Theorem 3.8).
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