Enumerative Combinatorics

by Stanley, Richard P.

• ISBN13: 9781107602625
• ISBN10: 1107602629
• Edition: 2nd
• Format: Paperback
• Copyright: 2011-12-12
• Publisher: Cambridge Univ Pr

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
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