Volume 14, Issue 4

We find limit distributions of the maximum size of a tree and of the number of trees of given size in an unlabelled random forest consisting of N rooted trees and n non-root vertices provided that N, n → ∞ so that 0 < C 1 ≤ N / √ n ≤ C 2 < ∞. With the use of these results, for the unlabelled graph of a random single-valued mapping of the set {1, 2, . . ., n } into itself we prove theorems on the limit behaviour of the maximum tree size and of the number of trees of size r as n → ∞ in the cases of fixed r and r / n 1/3 ≥ C 3 > 0.

We consider the number of cyclic vertices in a random single-valued mapping of a set of size n whose graph contains m cycles. We obtain a theorem that describes the limit behaviour of this characteristic as n → ∞, m / ln n → ∞, m / ln n = O (ln n ).

We describe the distributions of the number of ones, the number of runs of ones in a binary Markov sequence and their joint distribution. We find the generating functions of the distributions under consideration, calculate the means, variances and covariances. For these moments we give explicit and asymptotic formulas with estimates of accuracy. Formulas for the Gaussian approximations are also derived. We consider the corresponding operator equations. In this connection we describe a special model of random walks.

Let S be an arbitrary finite chain ring of prime characteristic. The aim of this paper is to describe the set of polynomial transformations and polynomial substitutions of S . The numbers of polynomial transformations and polynomial substitutions are found in some particular cases. We prove that if S is non-commutative, then any polynomial transformation of S is non-transitive.

We consider the problem on covering a non-oriented connected graph without loops and multiple edges by graphs of arbitrary finite bases. We introduce the notions of complexity of a covering, complexity of a graph and the Shannon function. In accordance with the asymptotic behaviour of the Shannon function, we introduce two classes of bases, almost dense and weakly dense bases. For the class of weakly dense bases and a special subclass of this class we suggest methods of constructing optimal exact coverings of a graph by bipartite basic graphs and find estimates of the complexity of such coverings. The suggested methods are based on algorithms for optimal exact covering of (0, 1)-matrices by arbitrary (0, 1)-matrices and on the connection between coverings of (0, 1)-matrices and coverings of a graph by bipartite graphs.

The method of boundary functionals was suggested by A. A. Sapozhenko for solving a series of enumeration problems. In this paper, we extend the method to the case of functional pairs with characteristic three and obtain an asymptotic formula for the sum of boundary functionals over regular families of subsets.

We consider periodic graphs of two types, chain and cyclic extension of an arbitrary graph. We give formulas which express the characteristic polynomials of such constructions in terms of the polynomials of the initial graphs, their subgraphs and the polynomials of a chain and a cycle. We present some applications of these results.