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# Enumerative Combinatorics: Volume 1

**by**Stanley, Richard P.

2nd

### 9781107602625

1107602629

Paperback

1/31/2012

Cambridge Univ Pr

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## Summary

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation and differential posets.

## Author Biography

Richard P. Stanley is a professor of applied mathematics at the Massachusetts Institute of Technology. He is universally recognized as a leading expert in the field of combinatorics and its applications to a variety of other mathematical disciplines. In addition to the seminal two-volume book Enumerative Combinatories, he is the author of Combinatories and Commutative Algebra (1983) and more than 100 research articles in National Academy of Sciences (elected in 1995), the 2001 Leroy P. Steele Prize for mathematical exposition, and the 2003 Schock Prize.

## Table of Contents

Preface | p. xi |

Acknowledgments | p. xiii |

What Is Enumerative Combinatorics? | p. 1 |

How to Count | p. 1 |

Sets and Multisets | p. 15 |

Cycles and Inversions | p. 22 |

Descents | p. 31 |

Geometric Representations of Permutations | p. 41 |

Alternating Permutations, Euler Numbers, and the cd-lndex of $$$n | p. 46 |

Permutations of Multisets | p. 54 |

Partition Identities | p. 61 |

The Twelvefold Way | p. 71 |

Two q-Analogues of Permutations | p. 80 |

Notes | p. 97 |

Bibliography | p. 100 |

Exercises for Chapter 1 | p. 103 |

Solutions to Exercises | p. 141 |

Sieve Methods | p. 195 |

Inclusion-Exclusion | p. 195 |

Examples and Special Cases | p. 198 |

Permutations with Restricted Position | p. 202 |

Ferrers Boards | p. 207 |

V-Partitions and Unimodal Sequences | p. 209 |

Involutions | p. 212 |

Determinants | p. 215 |

Notes | p. 218 |

Bibliography | p. 219 |

Exercises for Chapter 2 | p. 220 |

Solutions to Exercises | p. 231 |

Partially Ordered Sets | p. 241 |

Basic Concepts | p. 241 |

New Posets from Old | p. 246 |

Lattices | p. 248 |

Distributive Lattices | p. 252 |

Chains in Distributive Lattices | p. 256 |

Incidence Algebras | p. 261 |

The Möbius Inversion Formula | p. 264 |

Techniques for Computing Möbius Functions | p. 266 |

Lattices and Their Möbius Functions | p. 274 |

The Mobius Function of a Semimodular Lattice | p. 277 |

Hyperplane Arrangements | p. 280 |

Zeta Polynomials | p. 291 |

Rank Selection | p. 293 |

R-Labelings | p. 295 |

(P,¿)-Partitions | p. 298 |

Eulerian Posets | p. 310 |

The cd-Index of an Eulerian Poset | p. 315 |

Binomial Posets and Generating Functions | p. 320 |

An Application to Permutation Enumeration | p. 327 |

Promotion and Evacuation | p. 330 |

Differential Posets | p. 334 |

Notes | p. 345 |

Bibliography | p. 349 |

Exercises for Chapter 3 | p. 353 |

Solutions to Exercises | p. 408 |

Rational Generating Functions | p. 464 |

Rational Power Series in One Variable | p. 464 |

Further Ramifications | p. 467 |

Polynomials | p. 471 |

Quasipolynomials | p. 474 |

Linear Homogeneous Diophantine Equations | p. 475 |

Applications | p. 488 |

The Transfer-Matrix Method | p. 500 |

Notes | p. 523 |

Bibliography | p. 526 |

Exercises for Chapter 4 | p. 528 |

Solutions to Exercises | p. 548 |

Appendix: Graph Theory Terminology | p. 571 |

First Edition Numbering | p. 575 |

List of Notation (Partial) | p. 581 |

Index | p. 585 |

Table of Contents provided by Ingram. All Rights Reserved. |