Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some Properties and Combinatorial Implications of Weighted Small
  Schr\"oder Numbers   The $n^{\text{th}}$ small Schr\"oder number is $s(n) = \sum_{k \geq 0}
s(n,k)$, where $s(n,k)$ denotes the number of plane rooted trees with $n$
leaves and $k$ internal nodes that each has at least two children. In this
manuscript, we focus on the weighted small Schr\"oder numbers $s_d(n) = \sum_{k
\geq 0} s(n,k) d^k$, where $d$ is an arbitrary fixed real number. We provide
recursive and asymptotic formulas for $s_d(n)$, as well as some identities and
combinatorial interpretations for these numbers. We also establish connections
between $s_d(n)$ and several families of Dyck paths.

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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin