On a balanced property of derangements

Details On a balanced property of derangements On a balanced property of derangements

2006-06-12

arxiv.org/abs/math/0606277v1

Mathematics

Numerical Analysis

Scientific paper

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 $a$.

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