{'fraction_non_alphanumeric': 0.028930817610062894, 'fraction_numerical': 0.0012578616352201257, 'mean_word_length': 3.8536585365853657, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the distribution of the mth maximal cycle lengths of random A-permutations

Let S n be the symmetric group of all permutations of degree n, A be some subset of the set of natural numbers N , and T n = T n (A) be the set of all permutations of S n with cycle lengths belonging to A. The permutations of T n are called A-permutations. We consider a wide class of the sets A with the asymptotic density σ > 0. In this article, the limit distributions are obtained for μ m (n)/n as n → ∞ and m ∈ N is fixed. Here μm (n) is the length of the mth maximal cycle in a random permutation uniformly distributed on T n . It is shown here that these limit distributions coincide with the limit distributions of the corresponding functionals of the random permutations in the Ewens model with parameter σ.