3 research outputs found
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 -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 is chosen uniformly among all
permutations on elements, the events that has descents in a set
of positions, and that is a derangement, are positively correlated