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 $Pi = B_1/B_2/dots/B_k in P(n,k)$ in which the blocks are listed in increasing order of their least elements, let $pi = pi_1pi_2dotspi_n$ denote the canonical sequential form wherein $jin B_{nj}$ for all $jin [n]$. 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 $P(n, k).$