Niklas Eriksen ; Ragnar Freij ; Johan Wästlund

Enumeration of derangements with descents in prescribed positions
dmtcs:2738 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2738
Enumeration of derangements with descents in prescribed positions
Authors: Niklas Eriksen ^{1}; Ragnar Freij ^{1}; Johan Wästlund ^{2,}^{1}
NULL##NULL##NULL
Niklas Eriksen;Ragnar Freij;Johan Wästlund
1 Department of Mathematical Sciences
2 Department of Mathematics
We enumerate derangements with descents in prescribed positions. A generating function was given by GuoNiu 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.