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Advanced Combinatorics COMTET LOUIS COMTET University of Paris-&d (Orsay), France ADVANCED COMBINATORICS The Art of Finite and In_fiite Ex,namions 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. Nienhays TABLE OF CONTENTS Library of Congress Catalog Card Number 73-8609 1 INTKODUCTION 1X SYMBOLS AND ABRKEVIATlONS XI Cloth edition: ISBN 90 277 0380 Y Paperback edition: ISBN 90 277 0441 4 CHAPTER I. VOCABULARY OF COMBINATORIAL ANALYSlS I 1.1. Subsets of a Set; Operations 1 I .2. Product Sets 3 1.3. Maps 4 Published by D. Reidel Publishing Company, 1.4. Arrangements, Permutations 5 P.O. Box 17, Dordrecht, Holland 1.5. Combinations (without repetitions) or Blocks 7 t.6. Binomial Identity 12 Sold and distributed in the U.S.A., Canada, and Mexico 1.7. Combinations with Repetitions 15 by D. Reidel Publishing Company, Inc. 1.8. Subsets of [II], Random Walk 19 306 Dartmouth Street, Boston, 1.9. Subsets of Z/rzZ 23 Mass. 02116, U.S.A. 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 I. 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 In] 94 2.2. Generating Functions ofp(n) and P(n, nz) 96 All Rights Reserved 2.3. Conditional Partitions 98 Copyright 0 1974 by D. Reidel Publishing Company, Dordrecht, Holland 2.4. Ferrers Diagrams 99 No part of this book may be reproduced in any form, by print, photoprint, microfilm, 2.5. Special Identities; ‘Formal’ and ‘Combinatorial’ Proofs 103 or any other means, without written permission from the publisher 2.6. Partitions with Forbidden Summands; Denumerants 108 Printed in The Netherlands by D. Reidel, Dordrecht Supplement and Exercises 115 VI TABLE OF CONTENI’S ‘I‘ABl.li 01’ C’ONTEN’I‘S VII CHAPTER III. IDENTITIES AND EXPANSIONS I27 Supplement and Exeicises 219 3.1. Expansion of a Product of Sums; Abel Identity I 27 C’IIAP I‘ER VI. PI~RhlU’l‘A’I IONS 230 3.2. Product of Formal Series; Lcibniz Formula 130 6. I. The Symmetric Group 230 3.3. Bell Polynomials 133 6.2. Counting Prol~lems Relaled to Ikconiposition in Cycles; I&- 3.4. Substitution of One Formal Series into Another; E~ormuJa of turn lo Stirling Numbers of‘ the First Kind 233 Fag di Bruno 137 6.3. Multil~cr~nutatioiis 235 3.5. Logarithmic and Potential Polynomials I40 6.4. Inversions of a E’crmutntion of In] 236 3.6. Inversion Formulas and Matrix Calculus 143 6.5. Permutations by Number of Rises; Eulcrian Numbers 240 3.7. Fractionary iterates of Formal Series 144 6.6. Groups of Pcrniutalions; Cycle Indicak)r Polynomial; 13urn- 3.8. Inversion Formula of Lagrange 14s side ‘I’lieorem 246 I.51 3.9. Finite Summation Formulas 6.7. ‘l‘l1corem of PGlya 250 Supplement and Exercises 155 Supplcnienl aritl Excwises 254 CHAPTER IV. SIEVE FORMULAS I76 (:IIAI’l‘ER VII. EXAMPLES OF INI:QlJhLITlES AND ESl‘lMA’I‘I‘:S 268 4.1. Number of Elements of a Union or Inlersection I76 7. I. (.‘onvexity and Ilnimotlality ofC.‘ombinatorial Sequences 268 4.2. The ‘probl&me des renconlres’ 1 S) 7.2. Sperncr Systems 271 4.3. The ‘probkme des m&ages 182 7.3. Asymptofic Study of file Number of Regular C;raphs of Order 4.4. Boolean Algebra Generated by a System of Subsets IS5 ‘l‘wo on N 273 4.5. The Method of R&nyi for Linear Inequalities I SC) 7.1. Random Permulatinns 27Y 4.6. PoincarC Formula lOI ‘7.5. ‘1 heorcm of Rnmsq %X3 4.7. Bonferroni Inequalities IO3 ‘l.c>. Rinnr! (Bicolour) Ramsey Numbers 287 4.8. Formulas of Ch. Jordan I OS 7.7. Squa1-es in Rel:ltions 258 4.9. Permanents I ofl Supplement and Exercises 291 Suppiemeni and Exercises I ox I-~JNI~AMENTAI. NIJMERICAI. ‘I‘h131,ES 305 CHAPTER V. STIRLING NUMBERS 204 Factorials with Their Prime Factor Decomposition 20 5.1. Stirling Numbers of the Second Kind S(n, k) and Partitions of Binomial C’oefkicnls 306 Sets 701 Parlifions of Itilegers 307 5.2. Generating Functions for ,~(/I, k) ?Oc, Ikll Polynomials 30 7 5.3. Recurrence Relations between the S(n, 1~) 2.08 Iq~rit Iiriiic IWgn0mi:tls 308 5.4. The Number m(rt) of Partitions or Equivalence Relations of a 1’31 ti:llly Ordinary Ml p0lync,niinls 309 Set with II Elements 2 I C) Mllltinoniinl <‘oefi&nls 309 5.5. Stirling Numbers of the First Kind .T(II, k) and iheir (knrr:>tin,g Stirling N~~mbcrs ofllie 13lsl ICincl 310 Functions 31 2 Slil lirlg NIII~~IXYS ofille S~~c~c)wl !:.ilrcl :ind llxpon~n~i:~l Ntrn~lws 310 5.6. Recurrence Relations hetwecn !!w:c(!,, l:) .!! 4 31 .! 5.7. The Values of ,y(II, k) 2 I 6 5.8. Congruence Problems ?lR IrJIJr:y 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 voiume. 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- t.c;,,IGU--..--A ac^L a^ bGILLI,II--.A-:- III”luGlIC- ^_^^ A .GrLWlLll 1111,LGc-fc- ~LIL&LUIGJ,-L --.- 4-_--I ,ka”G1,^_.^ a b”III”IIIQL”IIcu---L:..,,,*:,.l character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathema- tics, but exclusively forfinite 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 SYMBOLS AND ABBREVIATIONS of compendium. The iirst 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 R(N) set of k-arrangements of N ( name and the year of publication. Thus, [Abel, 18261 refers, in the Bn,k partial Bell polynomials 1_,\ bibliography of articles, to the paper by Abel, published in 1826, Books are C set of complex numbers indicated by a star. So, for instance, [*Riordan, 19681 refers, in the biblio- E(X) expectation of random variable X r gruphy of books, to the book by Riordan, published in 1968. Suffixes a, b, GF generating function c, distinguish, for the same author, different articles that appeared in the N denotes, throughout the book, a finite set with n elements, IN] = n same year. N set of integers > 0 Each chapter is virtually independent of. the others, except of the P(A) probability of event A fist; but the use of the index will make it easy to consult each part of the WV set of subsets of N book separately. ‘9-W) set of nonemepty subsets of N I have taken the opportunity in this English edition to correct some %(N) set of subsets of N containing k elements printing errors and to improve certain points, taking into account the A+B = A v B, understanding that A n B = 8 suggestions which several readers kindly communicated to me and to R set of real numbers whom I feel indebted and most grateful. RV random variable Z set of all integers >cO n difference operator n indicates beginning and end of the proof of a theorem := equals by definition [nl the set (1,2, 3, . ., n} of the first n positive integers n! n factorial= the product i.2.3. n b>k =x(x-I)...@--k+l) Wk =x(x+1)...@!-k-1) bl the greatest integer less than or equal to x lbll the nearest integer to x (2 binomial coefficient = (n),/k! sh k) Stirling number of the first kind s(n, k) Stirling number of the second kind INI number of elements of set N F bound variable, with dot underneath CA, A complement of subset A Cmf coefficient of t” in the formal seriesf (x I P} set of all x with property B NM set of maps of M into N CHAPTER1 VOCABULARY OF COMBINATORIAL ANALYSIS In this chapter we define the language we will use and we introduce those elementary concepts which will be referred to throughout the book.
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