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  • Author: V. N. SACHKOV x
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Abstract

We consider the generating functions of the form exp{xg(t)}, where g(t) is a polynomial. These functions generate sequences of polynomials an(x), n = 0, 1,… Each polynomial g(t) is in correspondence with configurations of weight n whose sizes of components are bounded by the degree of the polynomial g(t). The polynomial an(x) is the generating function of the numbers ank, k = 1, 2,…, determining the number of configurations of weight n with k components.

We give asymptotic formulas as n → ∞ for the number of configurations of weight n and limit distributions for the number of components of a random configuration.

As an illustration we show how asymptotic formulas for the number of permutations and the number of partitions of a set with restriction on the cycle lengths and the sizes of blocks can be obtained with the use of the theory of configurations generated by polynomials. We obtain limit distributions of the number of cycles and the number of blocks of such random permutations and random partitions of sets.

Abstract

A square non-negative matrix is called primitive if all the elements of some power of this matrix are positive. The exponent of a primitive matrix is the minimum power satisfying this condition. The exponent of a class of primitive matrices is the minimum power such that all the matrices of the class to this power have positive elements only. In this paper, bounds for the exponents of primitive matrices and classes of matrices are obtained.