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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 $sigma$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $sigma$. 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 $sigma$, the probability of a random element being in a cycle longer than $gamma n$ approaches $(1-gamma)^sigma$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $sigma$. 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.
In this note we investigate correlation inequalities for `up-sets of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result s
Recall that an excedance of a permutation $pi$ is any position $i$ such that $pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted
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
We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $tau
We use moment method to understand the cycle structure of the composition of independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of small cycles