On a Recurrence involving Stirling Numbers

We count the number Z ( n ) of (not necessarily maximal) chains from 0 to 1 in the partition lattice of an n -set. This function satisfies the recurrence Z ( n ) = ∑ k − 1 n − 1 S ( n , k ) Z ( k ) where S ( n , k ) denotes the Stirling numbers of the second kind. We find the asymptotic order of magnitude of Z ( n ).