Enumeration of Derangements with Descents in Prescribed Positions

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, aswell 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 \p$ is a derangement, are positively correlated