  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 $p$.
