## Combinatorial Methods in Discrete MathematicsDiscrete mathematics is an important tool for the investigation of various models of functioning of technical devices, especially in the field of cybernetics. Here the author presents some complex problems of discrete mathematics in a simple and unified form using an original, general combinatorial scheme. Professor Sachkov's aim is to focus attention on results that illustrate the methods described. A distinctive aspect of the book is the large number of asymptotic formulae derived. Professor Sachkov begins with a discussion of block designs and Latin squares before proceeding to treat transversals, devoting much attention to enumerative problems. The main role in these problems is played by generating functions, considered in Chapter 4. The general combinatorial scheme is then introduced and in the last chapter Polya's enumerative theory is discussed. This is an important book for graduate students and professionals that describes many ideas not previously available in English; the author has updated the text and references where appropriate. |

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### Contents

1 Combinatorial configurations | 1 |

2 Transversals and permanents | 49 |

3 Generating functions | 102 |

4 Graphs and mappings | 165 |

5 The general combinatorial scheme | 209 |

6 Polyas theorem and its applications | 272 |

295 | |

303 | |

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### Common terms and phrases

allocations asymptotic formula automaton bijection binary relation called cells columns combinatorial scheme commutative asymmetric n-basis composition law condition conﬁgurations cycle class cycle index cycles of length cyclic deﬁned deﬁnition denote determined enumerator equation equivalence classes equivalence relation exists ﬁeld ﬁnd ﬁnite set ﬁrst formal power series G i G graph group G group of substitutions Hadamard matrix Hence it follows identical particles identity incidence matrix inequality integral inverse labeled Latin rectangle Lemma linear mapping matrix natural number non-commutative asymmetric n-basis non-negative number of m-samples number of partitions obtain one-to-one correspondence orthogonal Latin squares particles perA permutation matrices polynomials primary speciﬁcation probabilistic automaton Proof properties proved recurrence relation residue classes respect rooted trees saddle point satisﬁes secondary speciﬁcation sequence Stirling numbers subgroup subsets substitutions of degree summands symmetric group theorem total number transpositions transversal unrooted trees variables vector vertex zero