Records in set partitions
The electronic journal of combinatorics, Tome 17 (2010)
A partition of $[n]=\{1,2,\ldots,n\}$ is a decomposition of $[n]$ into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet, known as a restricted growth function. An element $a_i$ in such a word $\pi$ is a strong (weak) record if $a_i> a_j$ ($a_i\geq a_j$) for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all words corresponding to partitions of $[n]$, as well as for the sum of the positions of the records. In addition we find the asymptotic mean values and variances for the number, and for the sum of positions, of strong (weak) records in all partitions of $[n]$.
@article{10_37236_381,
author = {Arnold Knopfmacher and Toufik Mansour and Stephan Wagner},
title = {Records in set partitions},
journal = {The electronic journal of combinatorics},
year = {2010},
volume = {17},
doi = {10.37236/381},
zbl = {1193.05024},
url = {http://geodesic.mathdoc.fr/articles/10.37236/381/}
}
Arnold Knopfmacher; Toufik Mansour; Stephan Wagner. Records in set partitions. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/381
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