Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and
  Rooks   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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: hep-lat