### The largest singletons in weighted set partitions and its applications

*Yidong Sun, Yanjie Xu*

#### Abstract

Recently, Deutsch and Elizalde studied the largest fixed points of
permutations. Motivated by their work, we consider the analogous
problems in weighted set partitions. Let
A

_{n,k}(t) denote the total weight of partitions on [n+1]={1,2, ..., n+1} with the largest singleton {k+1}. In this paper, explicit formulas for A_{n,k}(t) and many combinatorial identities involving A_{n,k}(t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, eqnarray* ∑_{k=0}^{n}(n choose k)D_{k+1}(n+1)^{n-k}= n^{n+1}, where D_{k}is the number of permutations of [k] with no fixed points.Full Text: PDF PostScript