Counting Subwords in a Partition of a Set

Toufik Mansour, Mark Shattuck, Sherry H.F. Yan
2010 Electronic Journal of Combinatorics  
A partition $\pi$ of the set $[n]=\{1,\ldots,n\}$ is a collection $\{B_1,\ldots ,B_k\}$ of nonempty disjoint subsets of $[n]$ (called blocks) whose union equals $[n]$. In this paper, we find explicit formulas for the generating functions for the number of partitions of $[n]$ containing exactly $k$ blocks where $k$ is fixed according to the number of occurrences of a subword pattern $\tau$ for several classes of patterns, including all words of length 3. In addition, we find simple explicit
more » ... imple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of $[n]$ containing $k$ blocks, providing both algebraic and combinatorial proofs.
doi:10.37236/291 fatcat:o2cpqf7mlfcbxjxoabjl3zu57y