Enumeration of derangements with descents in prescribed positions

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We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $λ$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $π$ is chosen uniformly among all permutations on $n$ elements, the events that $π$ has descents in a set $S$ of positions, and that $π$ is a derangement, are positively correlated.

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Combinatorics, 05A05; 05A15

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http://salesiana.dossiersoluciones.com/handle/123456789/173957

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0811.1925
 http://arxiv.org/abs/0811.1925

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arXiv