CombinatoricsA mathematical gem–freshly cleaned and polished This book is intended to be used as the text for a first course in combinatorics. the text has been shaped by two goals, namely, to make complex mathematics accessible to students with a wide range of abilities, interests, and motivations; and to create a pedagogical tool, useful to the broad spectrum of instructors who bring a variety of perspectives and expectations to such a course. Features retained from the first edition:
Highlights of Second Edition enhancements:

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Contents
1  
Chapter 2 The Combinatorics of Finite Functions  117 
Chapter 3 Pólyas Theory of Enumeration  175 
Chapter 4 Generating Functions  253 
Chapter 5 Enumeration in Graphs  337 
Chapter 6 Codes and Designs  421 
Appendix A1 Symmetric Polynomials  477 
Appendix A2 Sorting Algorithms  485 
Appendix A3 Matrix Theory  495 
Bibliography  501 
Hints and Answers to Selected OddNumbered Exercises  503 
541  
547  
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Common terms and phrases
balls Bell numbers binary word binomial Cayley table chromatic polynomial closed formula codeword coefﬁcient color patterns column combinatorial Compute Conﬁrm Corollary corresponding cube cycle index cycle index polynomial cycle type deﬁned Deﬁnition degree sequence Denote dictionary order disjoint cycle factorization edges elementary symmetric functions elements Equation equivalent exactly Example Exercise exponential generating function Figure ﬁnd ﬁrst ﬁxed but arbitrary ﬁxed points follows G ¼ G Sm graph G group of degree Hadamard matrix Hint identity illustrated in Fig incidence matrix inequivalent integer isomorphic Lemma Let G linear code minimal symmetric polynomials modulo monomial multinomial multinomial theorem multiple nonisomorphic graphs nonnegative integer number of partitions obtained onetoone Pascal’s permutation group plane graph positive integer Prove rearranging recurrence Section Show squares of order Stirling numbers subgraph subgroup subsets Suppose symmetric polynomials threshold graph unique urns vertex vertices yields
Popular passages
Page xii  It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general the more difficult) the idea. Thus the idea of an "irrational" is deeper than that of an integer; and Pythagoras's theorem is, for that reason, deeper than Euclid's.
Page 20  It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. . . . The most important questions of life are, for the most part, really only problems of probability.