Enumerating set partitions according to the number of descents of size d or more

Let P(n, k) denote the set of partitions of {1, 2, ... , n} having exactly k blocks. In this paper, we find the generating function which counts the members of P(n, k) according to the number of descents of size d or more, where d ≥ 1 is fixed. An explicit expression in terms of Stirling numbersof the second kind may be given for the total number of such descents in all the members of P(n, k). We also compute the generating function for the statistics recording the number of ascents of size d or more and show that it has the same distribution on P(n, k) as the prior statistics for descents when d ≥ 2, by both algebraic and combinatorial arguments.