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In this paper, we consider a generalized m-excedance statistic on set partitions which is analogous to the usual excedance statistic on permutations when m = 0. We study it from a probabilistic perspective in which set partitions are regarded as geometrically distributed words satisfying a so-called restricted growth property. We derive a general set of recurrences satisfied by the relevant generating functions and, in the m = 0 and m = 1 cases, find an explicit formula for the distribution of the statistic recording the number of m-excedances in partitions having a fixed number of blocks. When m = 0, one can also determine the joint distribution for the number of excedances with the major index statistic on set partitions. Further recurrences may be given in this case as well as formulas for the mean number of excedances and the variance.
- Set partition
- Geometric random variable
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Cite this paper
Mansour, T & Shattuck, M. (2016). Set partitions and m-excedances. Notes on Number Theory and Discrete Mathematics, 22(1), 42-54.