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ADVANCED COMBINATORICS LOUIS COMTET University ofP aris-Sud (Orsay). France ADVANCED COMBINATORICS The Art of Finite and Infinite Expansions REVISED AND ENLARGED EDITION D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND / BOSTON-U.S.A. ANALYSE COMBINATOIRE, TOMES I ET II First published in 1970 by Presses Universitaires de France, Paris Translated from the French by J. W. Nienhuys Library of Congress Catalog Card Number 73-86091 ISBN-13: 978-94-010-2198-2 e-ISBN-13: 978-94-010-2196-8 DOl: 10.1007/978-94-010-2196-8 Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland Sold and distributed in the U.S.A., Canada, and Mexico by D. Reidel Publishing Company, Inc. 306 Dartmouth Street, Boston, Mass. 02116, U.S.A. All Rights Reserved Copyright © 1974 by D. Reidel Publishing Company, Dordrecht, Holland No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher TABLE OF CONTENTS INTRODUCTION IX SYMBOLS AND ABBREVIATIONS XI CHAPTER I. VOCABULARY OF COMBINATORIAL ANALYSIS 1 1.1. Subsets ofa Set; Operations 1 1.2. Product Sets 3 1.3. Maps 4 1.4. Arrangements, Permutations 5 1.5. Combinations (without repetitions) or Blocks 7 1.6. Binomial Identity 12 1.7. Combinations with Repetitions 15 1.8. Subsets of [n], Random Walk 19 1.9. Subsets of ZjnZ 23 1.10. Divisions and Partitions of a Set; Multinomial Identity 25 1.11. Bound Variables 30 1.12. Formal Series 36 1.13. Generating Functions 43 1.14. List of the Principal Generating Functions 48 1.15. Bracketing Problems 52 1.16. Relations 57 1.17. Graphs 60 1.18. Digraphs; Functions from a Finite Set into Itself 67 Supplement and Exercises 72 CHAPTER II. PARTITIONS OF INTEGERS 94 2.1. Definitions of Partitions of an Integer [n] 94 2.2. Generating Functions ofp(n) andP(n, m) 96 2.3. Conditional Partitions 98 2.4. Ferrers Diagrams 99 2.5. Specialldentities; 'Formal' and 'Combinatorial' Proofs 103 2.6. Partitions with Forbidden Summands; Denumerants 108 Supplement and Exercises 115 VI TABLE OF CONTENTS CHAPTER III. IDENTITIES AND EXPANSIONS 127 3.1. Expansion of a Product of Sums; Abel Identity 127 3.2. Product of Formal Series; Leibniz Formula 130 3.3. Bell Polynomials 133 3.4. Substitution of One Formal Series into Another; Formula of Faa di Bruno 137 3.5. Logarithmic and Potential Polynomials 140 3.6. Inversion Formulas and Matrix Calculus 143 3.7. Fractionary Iterates of Formal Series 144 3.8. Inversion Formula of Lagrange 148 3.9. Finite Summation Formulas 153 Supplement and Exercises 155 CHAPTER IV. SIEVE FORMULAS 176 4.1. Number of Elements ofa Union or Intersection 176 4.2. The 'probleme des rencontres' 180 4.3. The 'probleme des menages' 183 4.4. Boolean Algebra Generated by a System of Subsets 185 4.5. The Method of Renyi for Linear Inequalities 189 4.6. Poincare Formula 191 4.7. Bonferroni Inequalities 193 4.8. Formulas ofCh. Jordan 195 4.9. Permanents 196 Supplement and Exercises 198 CHAPTER V. STIRLING NUMBERS 204 5.1. Stirling Numbers of the Second Kind Sen, k) and Partitions of Sets 204 5.2. Generating Functions for Sen, k) 206 5.3. Recurrence Relations between the Sen, k) 208 5.4. The Number men) of Partitions or Equivalence Relations of a Set with n Elements 210 5.5. Stirling Numbers of the First Kind sen, k) and their Generating Functions 212 5.6. Recurrence Relations between the sen, k) 214 5.7. TheValuesofs(n,k) 216 5.8. Congruence Problems 218 T ABLE OF CONTENTS VII Supplement and Exercises 219 CHAPTER VI. PERMUTATIONS 230 6.1. The Symmetric Group 230 6.2. Counting Problems Related to Decomposition in Cycles; Re- turn to Stirling Numbers of the First Kind 233 6.3. Multipermutations 235 6.4. Inversions of a Permutation of en] 236 6.5. Permutations by Number of Rises; Eulerian Numbers 240 6.6. Groups of Permutations; Cycle Indicator Polynomial; Burn- side Theorem 246 6.7. Theorem ofP6lya 250 Supplement and Exercises 254 CHAPTER VII. EXAMPLES OF INEQUALITIES AND ESTIMATES 268 7.1. Convexity and Unimodality of Combinatorial Sequences 268 7.2. Sperner Systems 271 7.3. Asymptotic Study ofthe Number of Regular Graphs of Order TwoonN 273 7.4. Random Permutations 279 7.5. Theorem of Ramsey 283 7.6. Binary (Bicolour) Ramsey Numbers 287 7.7. Squares in Relations 288 Supplement and Exercises 291 FUNDAMENTAL NUMERICAL TABLES 305 Factorials with Their Prime Factor Decomposition 305 Binomial Coefficients 306 Partitions of Integers 307 Bell Polynomials 307 Logarithmic Polynomials 308 Partially Ordinary Bell polynomials 309 Multinomial Coefficients 309 Stirling Numbers of the First Kind 310 Stirling Numbers of the Second Kind and Exponential Numbers 310 BIBLIOGRAPHY 312 INDEX 337 INTRODUCTION Notwithstanding its title, the reader will not find in this book a systematic account of this huge subject. Certain classical aspects have been passed by, and the true title ought to be "Various questions of elementary combina torial analysis". For instance, we only touch upon the subject of graphs and configurations, but there exists a very extensive and good literature on this subject. For this we refer the reader to the bibliography at the end of the volume. The true beginnings of combinatorial analysis (also called combina tory analysis) coincide with the beginnings of probability theory in the 17th century. For about two centuries it vanished as an autonomous sub ject. But the advance of statistics, with an ever-increasing demand for configurations as well as the advent and development of computers, have, beyond doubt, contributed to reinstating this subject after such a long period of negligence. For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are con cerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathema tics, but exclusively forjinite sets. My idea is here to take the uninitiated reader along a path strewn with particular problems, and I can very well amagine that this journey may jolt a student who is used to easy generalizations, especially when only some of the questions I treat can be extended at all, and difficult or un solved extensions at that, too. Meanwhile, the treatise remains firmly elementary and almost no mathematics of advanced college level will be necessary. At the end of each chapter I provide statements in the form of exercises that serve as supplementary material, and I have indicated with a star those that seem most difficult. In this respect, I have attempted to write down x INTRODUCTION these 219 questions with their answers, so they can be consulted as a kind of compendium. The first items I should quote and recommend from the bibliography are the three great classical treatises of Netto, MacMahon and Riordan. The bibliographical references, all between brackets, indicate the author's name and the year of publication. Thus, [Abel, 1826] refers, in the bibliography of articles, to the paper by Abel, published in 1826. Books are indicated by a star. So, for instance, [*Riordan, 1968] refers, in the biblio graphy of books, to the book by Riordan, published in 1968. Suffixes a, b, c, distinguish, for the same author, different articles that appeared in the same year. Each chapter is virtually independent of the others, except of the first; but the use of the index will make it easy to consult each part of the book separately. I have taken the opportunity in this English edition to correct some printing errors and to improve certain points, taking into account the suggestions which several readers kindly communicated to me and to whom I feel indebted and most grateful. SYMBOLS AND ABBREVIATIONS m:iN) set of k-arrangements of N Bn•k partial Bell polynomials C set of complex numbers E(X) expectation of random variable X GF generating function N denotes, throughout the book, a finite set with n elements, INI = n N set of integers ~ 0 peA) probability of event A ~(N) set of subsets of N ~'(N) set of nonemepty subsets of N ~iN) set of subsets of N containing k elements A+B = A u B, understanding that A ('\ B = 0 R set of real numbers RV random variable Z set of all integers ~ 0 l::,. difference operator • indicates beginning and end of the proof of a theorem : = equals by definition en] the set {l, 2, 3, ... , n} of the first n positive integers n! n factorial = the product 1.2.3 ..... n (X)k =x(x-1) ... (x-k+ 1) (X)k =x(x+ 1) ... (x+k-1) [x] the greatest integer less than or equal to x IIxll the nearest integer to x (~) binomial coefficient = (n)k/k! s (n, k) Stirling number of the first kind Sen, k) Stirling number of the second kind INI number of elements of set N ~ bound variable, with dot underneath CA, A complementofsubsetA Ct n/ coefficient of tn in the formal series/ {x I .9!} set of all x with property .9! NM set of maps of Minto N