标题： 通过图和车棋的xD-斯特林数和拉赫数的推广 显示英文标题

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

摘要： 本文研究了与Weyl代数$W=\langle x,D|Dx-xD=1\rangle$中的正规顺序问题相关的两类Stirling数和Lah数的推广。 任何包含$\omega\in W$的词语，其中含有$m$ $x$ 和$n$ $D$，都可以表示为通常序形式$\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$，其中${{\omega}\brace {k}}$被称为词语$\omega$的第二类斯特林数。 本研究考虑了限制词$\omega$在$W$上关于序列$\{(xD)^{k}\}_{k\ge 0}$和$\{xD^{k}x^{k-1}\}_{k\ge 0}$的展开。 有趣的是，各个展开中的系数被证明是第一类斯特林数和拉赫数的推广。 这些系数将通过与词$\omega$相关的一些组合结构的计数来确定，包括准阈值图的递减森林分解和费勒板上的不攻击车放置。 扩展到$q$类比，还研究了在$q$变形的韦尔代数中的词的组合解释的加权细化。 显示英文摘要

摘要： 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.