Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The largest singletons in weighted set partitions and its applications   Recently, Deutsch and Elizalde studied the largest and the smallest fixed
points of permutations. Motivated by their work, we consider the analogous
problems in weighted set partitions. Let $A_{n,k}(\mathbf{t})$ denote the total
weight of partitions on $[n+1]$ with the largest singleton $\{k+1\}$. In this
paper, explicit formulas for $A_{n,k}(\mathbf{t})$ and many combinatorial
identities involving $A_{n,k}(\mathbf{t})$ are obtained by umbral operators and
combinatorial methods. As applications, we investigate three special cases such
as permutations, involutions and labeled forests. Particularly in the
permutation case, we derive a surprising identity analogous to the Riordan
identity related to tree enumerations, namely, \begin{eqnarray*}
\sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} where
$D_{k}$ is the $k$-th derangement number or the number of permutations of
$\{1,2,\dots, k\}$ with no fixed points.

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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex