The largest singletons of set partitions

Recently, Deutsch and Elizalde have studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let An,kAn,k denote the number of partitions of {1,2,…,n+1}{1,2,…,n+1} with the largest singleton {k+1}{k+1} for 0≤k≤n0≤k≤n. In this paper, several explicit formulas for An,kAn,k, involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods. Furthermore, many combinatorial identities involving An,kAn,k and Bell numbers are presented by operator methods, and congruence properties of An,kAn,k are also investigated. It is shown that the sequences (An+k,k)n≥0(An+k,k)n≥0 and (An+k,k)k≥0(An+k,k)k≥0 (mod pp) are periodic for any prime pp, and contain a string of p−1p−1 consecutive zeroes. Moreover their minimum periods are conjectured to be Np=pp−1p−1.