The book is devoted to the study of classical combinatorial structures such as random graphs, permutations, and systems of random linear equations in finite fields. The author shows how the application of the generalized scheme of allocation in the study of random graphs and permutations reduces the combinatorial problems to classical problems of probability theory on the summation of independent random variables. He offers recent research by Russian mathematicians, including a discussion of equations containing an unknown permutation, and the first English-language presentation of techniques for solving systems of random linear equations in finite fields. These new results will interest specialists in combinatorics and probability theory and will also be useful to researchers in applied areas of probabilistic combinatorics such as communication theory, cryptology, and mathematical genetics.
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algorithm allocating particles binomial distribution cells characteristic function conditions of Theorem connected graphs consider corresponding critical sets cycles of length deﬁned denote the number density difﬁcult Discrete Math distributed random variables distribution function distribution with parameter edges equal probabilities estimate exists a constant ﬁnd ﬁrst ﬁxed ﬁnite interval function F giant component hypercycles hypergraph identically distributed random independent identically distributed independent random variables integers labeled vertices Lemma limit distribution limit theorem logn matrix multinomial distribution nonnegative integers number of components number of cycles number of elements number of graphs obtain order statistics pennutations Poisson distribution probability tending proof of Theorem random forest random graph random mapping random permutation random variables $1 right-hand sides saddle-point method satisﬁes scheme of allocation Section sufﬁciently large summands summation systems of random take the values tends to zero total number unicyclic components uniformly V. F. Kolchin vector vertex