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Enumeration of derangements with descents in prescribed positions

Abstract : 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 $\lambda$-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 $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.
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Niklas Eriksen, Ragnar Freij, Johan Wästlund. Enumeration of derangements with descents in prescribed positions. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.385-396, ⟨10.46298/dmtcs.2738⟩. ⟨hal-01185430⟩

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