Refined Stirling Numbers: Enumeration of Special Sets of Permutations and Set-Partitions

The refined Stirling numbers of the first kindnm0m1 ··· mk−1 specify the number of permutations of n indices possessing mi cycles whose lengths modulo k are congruent to i, i=0,1,2,…,k−1. The refined Stirling numbers of the second kindnm0m1 ··· mk−1 are similarly defined in terms of set-partitions and the cardinalities of their disjoint blocks. Generating functions for these two types of refined Stirling numbers are derived using the Faà di Bruno formula. These generating functions allow the derivation of recurrence relations for both types of refined Stirling numbers.