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arXiv · 0907.5351

Profiles of permutations

Michael Lugo
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Abstract

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $σ$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $σ$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 \over 2} \log n + O(1)$, and the variance is of the same order. Furthermore, we show that in permutations of $[n]$ chosen from the Ewens distribution with parameter $σ$, the probability of a random element being in a cycle longer than $γn$ approaches $(1-γ)^σ$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $σ$. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.

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Michael Lugo. 2009-07-30. Profiles of permutations. https://arxiv.org/abs/0907.5351

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