## Introduction to CombinatoricsThe growth in digital devices, which require discrete formulation of problems, has revitalized the role of combinatorics, making it indispensable to computer science. Furthermore, the challenges of new technologies have led to its use in industrial processes, communications systems, electrical networks, organic chemical identification, coding theory, economics, and more. With a unique approach, Introduction to Combinatorics builds a foundation for problem-solving in any of these fields. Although combinatorics deals with finite collections of discrete objects, and as such differs from continuous mathematics, the two areas do interact. The author, therefore, does not hesitate to use methods drawn from continuous mathematics, and in fact shows readers the relevance of abstract, pure mathematics to real-world problems. The author has structured his chapters around concrete problems, and as he illustrates the solutions, the underlying theory emerges. His focus is on counting problems, beginning with the very straightforward and ending with the complicated problem of counting the number of different graphs with a given number of vertices. Its clear, accessible style and detailed solutions to many of the exercises, from routine to challenging, provided at the end of the book make Introduction to Combinatorics ideal for self-study as well as for structured coursework. |

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### Contents

Permutations and combinations | 1 |

The inclusionexclusion principle | 19 |

Partitions | 29 |

Stirlings approximation | 43 |

Partitions and generating functions | 58 |

Generating functions and recurrence relations | 80 |

Permutations and groups | 109 |

Group actions | 136 |

Graphs | 150 |

Counting patterns | 158 |

P6lyas Theorem | 167 |

Supplementary exercises | 194 |

Suggestions for further reading | 265 |

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### Common terms and phrases

algebraic asymptotic axes joining bijection binomial coefficients bridge hands Burnside's Theorem Cayley table chapter chessboard choices choose chosen combinatorial conjugacy classes contains corresponding cosets count the number counting problems cube cycle index cycle notation cycle type cycles of length defined diagram different graphs different patterns element of G example Exercise expression follows formula g e G G. H. Hardy given group action group G group of permutations Hardy-Ramanujan Hence identity element inequality integer inverse isomorphism labellings Lemma mapping mathematics matrix midpoints of opposite multiple natural numbers number of different number of elements number of partitions obtain orbits pack partition numbers pattern inventory pk(n polynomial function positive integer power series Prove qk(n Ramanujan recurrence relation riffle shuffle rotational symmetries solution of equation square Stirling's subgroup of G Subgroups of order suit distributions Suppose theory total number vertex vertices write