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Enumerative Combinatorics This is the second of a two-volume basic introduction to enumerative combinatorics at a level suitable for graduate students and research mathematicians. This volume covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course and focusing on combinatorics, especially the Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and rep- resentation theory. An appendix (written by Sergey Fomin) covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood- Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combina- torics to their work will find this an authoritative reference. Richard P. Stanley is Professor of Applied Mathematics at the Massachusetts Institute of Technology. He has held visiting positions at UCSD, the University of Strasbourg, California Institute of Technology, the University of Augsburg, Tokai University, and the Royal Institute of Technology in Stockholm. He has published over 100 research papers in algebraic combinatorics. In addition to the two-volume Enumerative Combinatorics, he has published one other book, Combinatorics and Commutative Algebra (Birkhauser; second edition, 1997). He is a fellow of the American Academy of Arts and Sciences, a member of the National Academy of Sciences, and a recipient of the Pblya Prize in Applied Combinatorics awarded by the Society for Industrial and Applied Mathematics.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 62 Already published J.L. Alperin Local representation theory Introduction to harmonic analysis on semisimple Lie groups P. Koosis The logarithmic integral !! J.F. Humphreys Reflection groups and Coxeter groups J.-P. Kahane Some random series of functions, 2nd edition J. Lambek & P.J. Scott Introduction to higher-order categorical logic P. Wojtaszczyk Banach spaces for analysts J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis 2 K. Petersen Ergodic theory P.T. Johnstone Stone spaces 3 4 W.H. Schikhof Ultrametric calculus 5 7 8 H. Matsumura Commutative ring theory 9 C.B. Thomas Characteristic classes and the cohomology of finite groups 10 M. Aschbacher Finite group theory 11 12 P. Koosis The logarithmic integral I 13 A. Pietsch Eigenvalues and s-numbers 14 S.J. Patterson An introduction to the theory of the Riemann zeta-function 15 H.J. Baues Algebraic homotopy 16 V.S. Varadarajan 17 W. Dicks & M. Dunwoody Groups acting on graphs 18 L.J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications 19 R. Fritsch & R. Piccinini Cellular structures in topology 20 H. Klingen Introductory lectures on Siegel modular forms 21 22 M.J. Collins Representations and characters of finite groups 24 H. Kunita Stochastic flows and stochastic differential equations 25 26 27 A. Frohlich & M.J. Taylor Algebraic number theory 28 K. Goebel & W.A. Kirk Topics in metric fixed point theory 29 30 D.J. Benson Representations and cohomology I 31 D.J. Benson Representations and cohomology II 32 C. Allday & V. Puppe Cohomological methods in transformation groups 33 C. Soule et al. Lectures on Arakelov geometry 34 A. Ambrosetti & G. Prodi A primer of nonlinear analysis 35 37 Y. Meyer Wavelets and operators 1 38 C. Weibel An Introduction to homological algebra 39 W. Bruns & J. Herzog Cohen-Macaulay rings 40 V. Snaith Explicit Brauer induction 41 G. Laumon Cohomology of Drinfeld modular varieties: Part I 42 E.B. Davies Spectral theory and differential operators 43 44 P. Mattila Geometry of sets and measures in Euclidean spaces 45 R. Pinsky Positive harmonic functions and diffusion 46 G. Ienenbaum Introduction to analytic and probabilistic number theory 47 C. Peskine Complex projective geometry 48 Y. Meyer & R. Coifman Wavelets 49 R. Stanley Enumerative combinatorics I 50 51 M. Audin 52 V. Jurdjevic Geometric control theory 53 H. Volklein Groups as Galois groups 54 55 D. Bump Automorphic forms and representations 56 G. Laumon Cohomology of Drinfeld modular varieties II 60 M.P. Brodmann & R.Y. Sharp Local cohomology J. Palis & F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations J. Diestel, H. Jarchow & A. Tonge Absolutely summing operators I. Porteous Clifford algebras and the classical groups Spinning tops J. Le Potier Lectures on vector bundles

ENUMERATIVE COMBINATORICS Volume 2 RICHARD P. STANLEY Massachusetts Institute of Technology AMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 First paperback edition 2001 Typeset in Times Roman 10/12 pt. in I9TEX 2E [TB] A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication data is available ISBN 0 521 56069 1 hardback ISBN 0 521 78987 7 paperback Transferred to digital printing 2004

Foreword Most textbooks written in our day have a short half-life. Published to meet the demands of a lucrative but volatile market, inspired by the table of contents of some out-of-print classic, garnished with multicolored tables, enhanced by nut- shell summaries, enriched by exercises of dubious applicability, they decorate the shelves of college bookstores come September. The leftovers after Registration Day will be shredded by Christmas, unwanted even by remainder bookstores. The pageant is repeated every year, with new textbooks on the same shelves by other authors (or a new edition if the author is the same), as similar to the preceding as one can make them, short of running into copyright problems. Every once in a long while, a textbook worthy of the name comes along; invari- ably, it is likely to prove aere perennius: Weber, Bertini, van der Waerden, Feller, Dunford and Schwartz, Ahlfors, Stanley. The mathematical community professes a snobbish distaste for expository writ- ing, but the facts are at variance with the words. In actual reality, the names of authors of the handful of successful textbooks written in this century are included in the list of the most celebrated mathematicians of our time. Only another textbook writer knows the pains and the endless effort that goes into this kind of writing. The amount of time that goes into drafting a satisfactory exposition is always underestimated by the reader. The time required to complete one single chapter exceeds the time required to publish a research paper. But far from wasting his or her time, the author of a successful textbook will be amply rewarded by a renown that will spill into the distant future. History is more likely to remember the name of the author of a definitive exposition than the names of many a research mathematician. I find it impossible to predict when Richard Stanley's two-volume exposition of combinatorics may be superseded. No one will dare try, let alone be able, to match the thoroughness of coverage, the care for detail, the definitiveness of proof, the elegance of presentation. Stanley's book possesses that rarest quality among textbooks: you can open it at any page and start reading with interest without having to hark back to page one for previous explanations. v

vi Foreword Combinatorics, which only thirty years ago was a fledgling among giants, may well be turning out to be a greater giant, thanks largely to Richard Stanley's work. Every one who deals with discrete mathematics, from category theorists to molecular biologists, owes him a large debt of gratitude. Gian-Carlo Rota March 21, 1998

Preface This is the second (and final) volume of a graduate-level introduction to enu- merative combinatorics. To those who have been waiting twelve years since the publication of Volume 1, I can only say that no one is more pleased to see Volume 2 finally completed than myself. I have tried to cover what I feel are the fundamen- tal topics in enumerative combinatorics, and the ones that are the most useful in applications outside of combinatorics. Though the book is primarily intended to be a textbook for graduate students and a resource for professional mathematicians, I hope that undergraduates and even bright high-school students will find something of interest. For instance, many of the 66 combinatorial interpretations of Catalan numbers provided by Exercise 6.19 should be accessible to undergraduates with a little knowledge of combinatorics. Much of the material in this book has never appeared before in textbook form. This is especially true of the treatment of symmetric functions in Chapter 7. Al- though the theory of symmetric functions and its connections with combinatorics is in my opinion one of the most beautiful topics in all of mathematics, it is a difficult subject for beginners to learn. The superb book by Macdonald on sym- metric functions is highly algebraic and eschews the fundamental combinatorial tool in this subject, viz., the Robinson-Schensted-Knuth algorithm. I hope that Chapter 7 adequately fills this gap in the mathematical literature. Chapter 7 should be regarded as only an introduction to the theory of symmetric functions, and not as a comprehensive treatment. As in Volume 1, the exercises play a vital role in developing the subject. If in reading the text the reader is left with the feeling of "what's it good for?" and is not satisfied with the applications presented there, then (s)he should turn to the exercises. Thanks to the wonders of electronic word processing, I found it much easier than for Volume 1 to assemble a wide variety of exercises and solutions. I am grateful to the many persons who have contributed in a number of ways to the improvement of this book. Special thanks go to Sergey Fomin for his many suggestions related to Chapter 7, and especially for his masterful exposition of the difficult material of Appendix 1. Other persons who have carefully read por- tions of earlier versions of the book and who have offered valuable suggestions vii

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