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  1. Home > Articles & Issues >
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  3. Enumeration of deran ...
Conference paper

Enumeration of derangements with descents in prescribed positions

Niklas Eriksen (1), Ragnar Freij (1), Johan Wästlund ORCID (2) (1)
(1) Department of Mathematical Sciences
(2) Department of Mathematics
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Imported on
January 31, 2017
Published on
December 31, 2008
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
Proceedings
DOI
10.46298/dmtcs.2738
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https://about.hal.science/hal-authorisation-v1
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Enumeration of derangements with descents in prescribed positions

Niklas Eriksen (1), Ragnar Freij (1), Johan Wästlund ORCID (2) (1)
(1) Department of Mathematical Sciences
(2) Department of Mathematics
Abstract
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.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • fixed point
  • Permutation statistic
  • descent
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