Enumeration of derangements with descents inprescribed positions

摘要

We enumerate derangements with descents in prescribed positions. A generatingfunction was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorialproof of this result, and derive several explicit formulas. To this end, we considerfixed point A-coloured permutations, which are easily enumerated. Several formulaeregarding these numbers are given, as well as a generalisation of Euler's differencetables. We also prove that except in a trivial special case, if a permutation 7r is chosenuniformly among all permutations on n elements, the events that 71 has descents ina set S of positions, and that π is a derangement, are positively correlated.