Asymptotic formulas and limit distributions for combinatorial configurations generated by polynomials
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V. N. Sachkov
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.
© de Gruyter
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- Asymptotic formulas and limit distributions for combinatorial configurations generated by polynomials
- On homomorphisms of many-sorted algebraic systems in connection with cryptographic applications
- A general approach to studying the stability of a Pareto optimal solution of a vector integer linear programming problem
- An upper bound for the number of maximal independent sets in a graph
- On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other
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Articles in the same Issue
- Asymptotic formulas and limit distributions for combinatorial configurations generated by polynomials
- On homomorphisms of many-sorted algebraic systems in connection with cryptographic applications
- A general approach to studying the stability of a Pareto optimal solution of a vector integer linear programming problem
- An upper bound for the number of maximal independent sets in a graph
- On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other
- One automaton model in biology
- On an automaton model of pursuit
