  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.
