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  • Author: YU. I. MEDVEDEV x
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We suggest a new approach to the study of the structural properties of random permutations based on defining some general parametric measure on the set S n of all permutations of degree n which assigns to a permutation with cyclic structure a = (a 1, . . . ,a n) the probability proportional to , where θ = (θ 1, . . . , θ n) is a parameter of the measure.

Abstract

We consider monic (with higher coefficient 1) polynomials of fixed degree n over an arbitrary finite field GF(q), where q ≥ 2 is a prime number or a power of a prime number. It is assumed that on the set Fn ={ƒn} of all qn such polynomials the uniform measure is defined which assigns the probability q-n to each polynomial. For an arbitrary polynomial ƒnFn, its local structure Kn = K(ƒn) is defined as the set of multiplicities of all irreducible factors in the canonical decomposition of ƒn, and various structural characteristics of a polynomial (its exact and asymptotic as n → ∞ distributions) which are functionals of Kn are studied. Directions of possible further research are suggested.