
SenPeng Eu

TungShan Fu

YuChang Liang

TsaiLien Wong
Keywords:
Stirling number, Nah number, Normal ordering problem, QuasiThreshold graphs, Rook numbers
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 $W=\langle x,DDxxD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{mn}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, 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 $\{xD^{k}x^{k1}\}_{k\ge 0}$. 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 quasithreshold graphs and nonattacking 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.