Enumeration of derangements with descents in prescribed positions
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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.
@article{10_37236_121,
author = {Niklas Eriksen and Ragnar Freij and Johan W\"astlund},
title = {Enumeration of derangements with descents in prescribed positions},
journal = {The electronic journal of combinatorics},
year = {2009},
volume = {16},
number = {1},
doi = {10.37236/121},
zbl = {1181.05003},
url = {http://geodesic.mathdoc.fr/articles/10.37236/121/}
}
TY - JOUR AU - Niklas Eriksen AU - Ragnar Freij AU - Johan Wästlund TI - Enumeration of derangements with descents in prescribed positions JO - The electronic journal of combinatorics PY - 2009 VL - 16 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.37236/121/ DO - 10.37236/121 ID - 10_37236_121 ER -
Niklas Eriksen; Ragnar Freij; Johan Wästlund. Enumeration of derangements with descents in prescribed positions. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/121
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