Lengyel's Constant

Let S be a set with n elements. The set of all subsets of S has elements. By a partition of S we mean a disjoint set of nonempty subsets (called blocks) whose union is S. The set of all partitions of S has elements, where is the nth Bell number and where denotes the Stirling number of the second kind. For example, We have recurrences ([1,2,3]) and asymptotics where is defined by. If U and V are subsets of S, write U<V if U is a proper subset of V. This endows the set of all subsets of S with a partial ordering; in fact, it is a lattice with maximum element S and minimum element. The number of chains of length k is. Hence the number of all chains from to S is ([1,3,4])