Geodesic
Counting subwords in a partition of a set
The electronic journal of combinatorics, Tome 17 (2010)
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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 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
Classification : 05A18, 05A15, 05A05, 68R05
Mots-clés : set partition, generating function, subword pattern 
@article{10_37236_291,
     author = {Toufik Mansour and Mark Shattuck and Sherry H.F. Yan},
     title = {Counting subwords in a partition of a set},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/291},
     zbl = {1193.05025},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/291/}
}
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Toufik Mansour; Mark Shattuck; Sherry H.F. Yan. Counting subwords in a partition of a set. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/291

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