Enumerating set partitions by the number of positions between adjacent occurrences of a letter

A emph{partition} $Pi$ of the set $[n]={1,2,ldots,n}$ is acollection ${B_1,ldots ,B_k}$ of nonempty disjoint subsets of$[n]$ (called emph{blocks}) whose union equals $[n].$ Supposethat the subsets $B_i$ are listed in increasing order of theirminimal elements and $pi=pi_1pi_2cdotspi_n$ denotes thecanonical sequential form of a partition of $[n]$ in which $i inB_{pi_i}$ for each $i.$ In this paper, we study the generatingfunctions corresponding to statistics on the set of partitions of$[n]$ with $k$ blocks which record the total number of positionsof $pi$ between adjacent occurrences of a letter. Among ourresults are explicit formulas for the total value of thestatistics over all the partitions in question, for which weprovide both algebraic and combinatorial proofs. In addition, wesupply asymptotic estimates of these formulas, the proofs of whichentail approximating the size of certain sums involving theStirling numbers. Finally, we obtain comparable results forstatistics on partitions which record the total number ofpositions of $pi$ of the same letter lying between two letterswhich are strictly larger