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 lambda-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, aswell 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 \p$ is a derangement, are positively correlated.

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Natural Sciences  (hsv)

Mathematics  (hsv)

Discrete Mathematics  (hsv)

Naturvetenskap  (hsv)

Matematik  (hsv)

Diskret matematik  (hsv)

Matematik  (oru)

Mathematics  (oru)