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Register a new userThe number of cycles of specified normalized length in permutations
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Authors
Michael Lugo
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We compute the limiting distribution, as n approaches infinity, of the number of cycles of length between gamma n and delta n in a permutation of [n] chosen uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma < delta <= 1/k for some integer k. This distribution is supported on {0, 1, ... k} and has 0th, 1st, ..., kth moments equal to those of a Poisson distribution with parameter log (delta/gamma). For more general choices of gamma, delta we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hypercubes with half-spaces; these integrals are analytically intractable but a recurrence specifying them can be given. The results herein provide a basis of comparison for similar statistics on restricted classes of permutations.
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We prove that the number of Hamilton cycles in the random graph G(n,p) is n!p^n(1+o(1))^n a.a.s., provided that pgeq (ln n+ln ln n+omega(1))/n. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e)^n(1+o(1))^n Hamilton cycles a.a.s.
Let $mathbb{F}_q$ be a finite field of odd characteristic. We study Redei functions that induce permutations over $mathbb{P}^1(mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $jgeq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Redei permutation of this type to exist over $mathbb{P}^1(mathbb{F}_q)$, characterize Redei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Redei involutions based on the number of fixed points, and procedures to construct Redei permutations with a prescribed number of fixed points and $j$-cycles for $j in {3,4,5}$.
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