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Real zeros and partitions without singleton blocks. (English) Zbl 1321.05014

Summary: 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 and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks. Also, we present a quick way to prove the corresponding statement for cycles of permutations in which each cycle is longer than a given integer \(r\).

MSC:

05A18 Partitions of sets
05C15 Coloring of graphs and hypergraphs
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