# Enumeration of derangements with descents in prescribed positions Journal article, 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 $\lambda$-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.

descent

fixed point

Permutation statistic

## Author

#### Niklas Eriksen

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

#### Ragnar Freij

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

#### Johan Wästlund

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

#### Electronic Journal of Combinatorics

1097-1440 (ISSN) 1077-8926 (eISSN)

Vol. 16 1 R32-

#### Subject Categories

Discrete Mathematics