Real Zeros and Partitions without singleton blocks

Details Real Zeros and Partitions without singleton blocks Real Zeros and Partitions without singleton blocks

2007-05-18

arxiv.org/abs/0705.2734v2

Mathematics

Combinatorics

13 pages

Scientific paper

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only. We apply this information to find the most likely number of blocks. As another application of the real zeros result, we prove that the number of blocks is normally distributed in such partitions. We present a quick way to prove the corresponding statement for cycles of permutations in which each cycle is longer than a given integer $r$.

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