Asymptotic formulas and limit distributions for combinatorial configurations generated by polynomials

Abstract

We consider the generating functions of the form exp{ xg ( t )}, where g ( t ) is a polynomial. These functions generate sequences of polynomials a n ( 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 a n ( x ) is the generating function of the numbers a nk , 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.