Geodesic
Limit behavior of the order statistics on the cycle lengths of random $A$-permutations
Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 148-160

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We consider a random permutation $\tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). Let $\zeta_n$ be the total number of cycles, and let $\eta_n(1)\leq\eta_n(2)\leq\dots\leq\eta_n(\zeta_n)$ be the ordered sample of cycle lengths of the permutation $\tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $\eta_n(m)$ with numbers $m$ in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $\eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340–357] who considered the case $A=\mathbf N$.
Keywords: random $A$-permutation, ordered sample for cycle length of a permutation, order statistics. 
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     author = {A. L. Yakymiv},
     title = {Limit behavior of the order statistics on the cycle lengths of random $A$-permutations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {148--160},
     publisher = {mathdoc},
     volume = {69},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/}
}
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A. L. Yakymiv. Limit behavior of the order statistics on the cycle lengths of random $A$-permutations. Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 148-160. http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/