Enumerating finite set partitions according to the number of connectors

Let P(n,k) denote the set of partitions of [n]={1,2,⋯,n} containing exactly k blocks. Given a partition ∏=B 1 /B 2 /⋯/B k ∈P(n,k) in which the blocks are listed in increasing order of their least elements, let π=π 1 π 2 ⋯π n denote the canonical sequential form wherein j∈B n j for all j∈[n]. In this paper, we supply an explicit formula for the generating function which counts the elements of P(n,k) according to the number of strings k1 and r(r+1), taken jointly, occurring in the corresponding canonical sequential forms. A comparable formula for the statistics on P(n,k) recording the number of strings 1k and r(r-1) is also given which may be extended to strings r(r-1)⋯(r-m) of arbitrary length using linear algebra. In addition, we supply algebraic and combinatorial proofs of explicit formulas for the total number of occurrences of k1 and r(r+1) within all the members of P(n,k).