In this note, by the umbra calculus method, the Sun and Zagiers congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
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 t
he largest singleton ${k+1}$ for $0leq kleq n$. 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 $(A_{n+k,k})_{ngeq 0}$ and $(A_{n+k,k})_{kgeq 0}$ (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 $p$.
