Refined stirling numbers: enumeration of special sets of permutations and set-partitions

  • Authors:

  • J. Katriel

  • Affiliations:

  • Department of Chemistry, Technion - Israel Institute of Technology, Haifa 32000, Israel

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2002

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Abstract

The refined Stirling numbers of the first kind [n m0 m1 ..... 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 kind {n m0 m1 ..... 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.