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# Enumeration of derangements with descents in prescribed positions

Niklas Eriksen (Institutionen för matematiska vetenskaper, matematik) ; Ragnar Freij (Institutionen för matematiska vetenskaper) ; Johan Wästlund (Institutionen för matematiska vetenskaper, matematik)
Electronic Journal of Combinatorics (1077-8926). Vol. 16 (2009), 1, p. R32.

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.

Nyckelord: Permutation statistic, fixed point, descent

CPL Pubid: 94521

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# Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)
Institutionen för matematiska vetenskaperInstitutionen för matematiska vetenskaper (GU)