Enumeration of Derangements with Descents in Prescribed Positions - Eriksen, Niklas; Freij, Ragnar; Wästlund, Johan (The Electronic Journal of Combinatorics, 2009)
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.