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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 denote the total
weight of partitions on with the largest singleton . In this
paper, explicit formulas for and many combinatorial
identities involving 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
is the -th derangement number or the number of permutations of
with no fixed points.Comment: 15page
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