Geodesic
On a balanced property of derangements
The electronic journal of combinatorics, Tome 13 (2006)
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We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of cycles of a randomly selected derangement of length $n$, then the probability that $k$ is congruent to a given $r$ modulo a given $q$ converges to $1/q$. Finally, we generalize our results to $a$-derangements, which are permutations in which each cycle is longer than $a$.
DOI : 10.37236/1128
Classification : 05A16
Mots-clés : generating polynomials, number of cycles 
@article{10_37236_1128,
     author = {Mikl\'os B\'ona},
     title = {On a balanced property of derangements},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1128},
     zbl = {1112.05008},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1128/}
}
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%J The electronic journal of combinatorics
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Miklós Bóna. On a balanced property of derangements. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1128

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