The largest singletons of set partitions

Abstract: Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let $A_{n,k}$ denote the number of partitions of ${1,2,\dots, n+1}$ with the largest singleton ${k+1}$ for $0\leq k\leq n$. In this paper, several explicit formulas for $A_{n,k}$, involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving $A_{n,k}$ and Bell numbers are presented by operator methods, and congruence properties of $A_{n,k}$ are also investigated. It will been showed that the sequences $(A_{n+k,k}){n\geq 0}$ and $(A{n+k,k})_{k\geq 0}$ (mod $p$) are periodic for any prime $p$, and contain a string of $p-1$ consecutive zeroes. Moreover their minimum periods are conjectured to be $N_p=\frac{p^p-1}{p-1}$ for any prime $p$.