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LetS   n    $ \mathfrak S_n $    be a semigroup of all mappings from the  n -element set  X  into itself. We consider a setS   n   (  A  )   $ \mathfrak S_n(A) $    of mappings fromS   n    $ \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 onS   n   (  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 &gt;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 &gt;0 and  ρ &gt;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.

Size distribution of the largest component of a random A -mapping