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=0ⁿ(n choose k)D_k+1(n+1)^n-k = nⁿ⁺¹, where D_k is the number of permutations of [k] with no fixed points.