Arsen L. Yakymiv
April 7, 2021
Let Sn$ \mathfrak S_n $ be a semigroup of all mappings from the n -element set X into itself. We consider a set Sn(A)$ \mathfrak S_n(A) $ of mappings from Sn$ \mathfrak S_n $ such that their contour sizes belong to the set A ⊆ N . These mappings are called A -mappings. Let a random mapping τ n have a distribution on Sn(A)$ \mathfrak S_n(A) $ such that each connected component with volume i ∈ N have weight ϑ i ⩾0. Let D be a subset of N . It is assumed that ϑ i → ϑ >0 for i ∈ D and ϑ i → 0 for i ∈ N ∖ D as i → ∞. Let μ ( n ) be the maximal volume of components of the random mapping τ n . We suppose that sets A and D have asymptotic densities ϱ >0 and ρ >0 in N respectively. It is shown that the random variables μ ( n )/ n converge weakly to random variable ν as n → ∞. The distribution of ν coincides with the limit distribution of the corresponding characteristic in the Ewens sampling formula for random permutation with the parameter ρ ϱ ϑ /2.