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Combinatorics: The Art of Counting
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Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020.
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Combinatorics: The Art of Counting
Bruce E. Sagan
GRADUATE STUDIESIN MATHEMATICS 210
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Marco GualtieriBjorn Poonen
Gigliola Staffilani (Chair)Jeff A. Viaclovsky
2020 Mathematics Subject Classification. Primary 05-01; Secondary 06-01.
For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-210
Library of Congress Cataloging-in-Publication Data
Names: Sagan, Bruce Eli, 1954- author.Title: Combinatorics : the art of counting / Bruce E. Sagan.Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Gradu-
ate studies in mathematics, 1065-7339 ; 210 | Includes bibliographical references and index. |Identifiers: LCCN 2020025345 | ISBN 9781470460327 (paperback) | ISBN 9781470462802 (ebook)Subjects: LCSH: Combinatorial analysisโTextbooks. | AMS: Combinatorics โ Instructional ex-
position (textbooks, tutorial papers, etc.). | Order, lattices, ordered algebraic structures โInstructional exposition (textbooks, tutorial papers, etc.).
Classification: LCC QA164 .S24 2020 | DDC 511/.6โdc23LC record available at https://lccn.loc.gov/2020025345
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cยฉ 2020 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept those granted to the United States Government.
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To Sally, for her love and support
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Contents
Preface xi
List of Notation xiii
Chapter 1. Basic Counting 1ยง1.1. The Sum and Product Rules for sets 1ยง1.2. Permutations and words 4ยง1.3. Combinations and subsets 5ยง1.4. Set partitions 10ยง1.5. Permutations by cycle structure 11ยง1.6. Integer partitions 13ยง1.7. Compositions 16ยง1.8. The twelvefold way 17ยง1.9. Graphs and digraphs 18ยง1.10. Trees 22ยง1.11. Lattice paths 25ยง1.12. Pattern avoidance 28Exercises 33
Chapter 2. Counting with Signs 41ยง2.1. The Principle of Inclusion and Exclusion 41ยง2.2. Sign-reversing involutions 44ยง2.3. The GarsiaโMilne Involution Principle 49ยง2.4. The Reflection Principle 52
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viii Contents
ยง2.5. The LindstrรถmโGesselโViennot Lemma 55ยง2.6. The Matrix-Tree Theorem 59Exercises 64
Chapter 3. Counting with Ordinary Generating Functions 71ยง3.1. Generating polynomials 71ยง3.2. Statistics and ๐-analogues 74ยง3.3. The algebra of formal power series 81ยง3.4. The Sum and Product Rules for ogfs 86ยง3.5. Revisiting integer partitions 89ยง3.6. Recurrence relations and generating functions 92ยง3.7. Rational generating functions and linear recursions 96ยง3.8. Chromatic polynomials 99ยง3.9. Combinatorial reciprocity 106Exercises 109
Chapter 4. Counting with Exponential Generating Functions 117ยง4.1. First examples 117ยง4.2. Generating functions for Eulerian polynomials 121ยง4.3. Labeled structures 124ยง4.4. The Sum and Product Rules for egfs 128ยง4.5. The Exponential Formula 131Exercises 134
Chapter 5. Counting with Partially Ordered Sets 139ยง5.1. Basic properties of partially ordered sets 139ยง5.2. Chains, antichains, and operations on posets 145ยง5.3. Lattices 148ยง5.4. The Mรถbius function of a poset 154ยง5.5. The Mรถbius Inversion Theorem 157ยง5.6. Characteristic polynomials 164ยง5.7. Quotients of posets 168ยง5.8. Computing the Mรถbius function 174ยง5.9. Binomial posets 178Exercises 183
Chapter 6. Counting with Group Actions 189ยง6.1. Groups acting on sets 189ยง6.2. Burnsideโs Lemma 192ยง6.3. The cycle index 197
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Contents ix
ยง6.4. RedfieldโPรณlya theory 200ยง6.5. An application to proving congruences 205ยง6.6. The cyclic sieving phenomenon 209Exercises 213
Chapter 7. Counting with Symmetric Functions 219ยง7.1. The algebra of symmetric functions, Sym 219ยง7.2. The Schur basis of Sym 224ยง7.3. Hooklengths 230ยง7.4. ๐-partitions 235ยง7.5. The RobinsonโSchenstedโKnuth correspondence 240ยง7.6. Longest increasing and decreasing subsequences 244ยง7.7. Differential posets 248ยง7.8. The chromatic symmetric function 253ยง7.9. Cyclic sieving redux 256Exercises 259
Chapter 8. Counting with Quasisymmetric Functions 267ยง8.1. The algebra of quasisymmetric functions, QSym 267ยง8.2. Reverse ๐-partitions 270ยง8.3. Chain enumeration in posets 274ยง8.4. Pattern avoidance and quasisymmetric functions 276ยง8.5. The chromatic quasisymmetric function 279Exercises 283
Appendix. Introduction to Representation Theory 287ยงA.1. Basic notions 287Exercises 292
Bibliography 293
Index 297
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Preface
Enumerative combinatorics has seen an explosive growth over the last 50 years. Thepurpose of this text is to give a gentle introduction to this exciting area of research. So,rather than trying to cover many different topics, I have chosen to give a more leisurelytreatment of some of the highlights of the field. My goal has been towrite the expositionso it could be read by a student at the advanced undergraduate or beginning graduatelevel, either as part of a course or for independent study. The reader will find it similarin tone to my book on the symmetric group. I have tried to keep the prerequisites to aminimum, assuming only basic courses in linear and abstract algebra as background.Certain recurring themes are emphasized, for example, the existence of sum and prod-uct rules first for sets, then for ordinary generating functions, and finally in the case ofexponential generating functions. I have also included some recent material from theresearch literature which, to my knowledge, has not appeared in book form previously,such as the theory of quotient posets and the connection between pattern avoidanceand quasisymmetric functions.
Most of the exercises should be doable with a reasonable amount of effort. A fewunsolved conjectures have been included among the problems in the hope that an in-terested studentmight wish to tackle one of them. They are, of course, marked as such.
A few words about the title are in order. It is in part meant to be a tip of the hat toDonald Knuthโs influential series of books The art of computer programing, Volumes1โ3 [51โ53], which, amongmany other things, helped give birth to the study of patternavoidance through its connection with stack sorting; see Exercise 36 in Chapter 1. Ihope that the title also conveys some of the beauty found in this area of mathemat-ics, for example, the elegance of the Hook Formula (equation (7.10)) for the numberof standard Young tableaux. In addition I should mention that, due to my own pref-erences, this book concentrates on the enumerative side of combinatorics and mostlyignores the important extremal and existential parts of the field. The reader interestedin these areas can consult the books of Flajolet and Sedgewick [25] and of van Lint [95].
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xii Preface
This book grew out of the lecture notes which I have compiled over years of teach-ing the graduate combinatorics course at Michigan State University. I would like tothank the students in these classes for all the feedback they have given me about thevarious topics and their presentation. I am also indebted to the following colleagues,some of whom taught from a preliminary version of this book, who provided me withsuggestions aswell as catching numerous typographical errors: Matthias Beck,MoussaBenoumhani, Andreas Blass, Seth Chaiken, Sylvie Corteel, Georges Grekos, RichardHensh, Nadia Lafreniรจre, Duncan Levear, and Tom Zaslavsky. Darij Grinberg deservesspecial mention for providing copious comments and corrections as well as providinga number of interesting exercises. I also received valuable feedback from four anony-mous referees. Finally, I wish to express my appreciation of Ina Mette, my editor atthe American Mathematical Society. Without her gentle support and persistence, thistext would never have seen the light of day. Because I typeset this document myself,all errors can be blamed on my computer.
East Lansing, Michigan, 2020
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List of Notation
Symbol Definition Page
๐ด(๐ท) arc set of digraph ๐ท 21๐ด(๐บ) adjacency matrix of graph ๐บ 60๐(๐บ) set of acyclic orientations of ๐บ 103๐(๐บ) number of acyclic orientations of ๐บ 103๐ด([๐], ๐) set of permutations ๐ in ๐๐ having ๐ descents 121๐ด(๐, ๐) Eulerian number, cardinality of ๐ด([๐], ๐) 121๐ด๐(๐) Eulerian polynomial 122๐(๐) atom set of poset ๐ 169Asc ๐ ascent set of a proper coloring ๐ 279asc ๐ ascent number of a proper coloring ๐ 279Asc๐ ascent set of permutation ๐ 76asc ๐ ascent number of permutation ๐ 76Av๐(๐) the set of permutations in ๐๐ avoiding ๐ 29๐ผ๐ reversal of composition ๐ผ 32๏ฟฝฬ๏ฟฝ expansion of composition ๐ผ 274๐ผ(๐ถ) rank composition of chain ๐ถ 275๐ต(๐บ) incidence matrix of graph ๐บ 61๐ต(๐) set of partitions of the set ๐ 10๐ต๐ Boolean algebra on [๐] 140๐ตโ poset of subsets of โ 178๐ต(๐) ๐th Bell number 10
xiii
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xiv List of Notation
Symbol Definition Page
โ complex numbers 1๐๐(๐) number of cycles of length ๐ in group element ๐ 197๐ถ๐ฟ๐ claw poset with ๐ atoms 169co ๐ content of tableau ๐ 225๐ถ๐ cycle with ๐ vertices 19๐ถ๐ chain poset of length ๐ 139๐๐ฅ(๐) column insertion of element ๐ฅ into tableau ๐ 245๐ถโ chain poset on โ 178๐ถ(๐) Catalan number 26๐([๐], ๐) set of permutations in ๐๐ with ๐ cycles 12๐(๐, ๐) signless Stirling number of the first kind 12๐๐(๐ฟ, ๐) ordered ๐ cycle decompositions of permutations of ๐ฟ 127โ๐ vector space generated by set ๐ over โ 248โ[๐ฅ] polynomial algebra in ๐ฅ over โ 71โ[[๐ฅ]] formal power series algebra in ๐ฅ over โ 81๐(๐) set of functions compatible with ๐ 236๐๐(๐) set of functions compatible with ๐ bounded by๐ 236Des ๐ descent set of tableau ๐ 271Des๐ descent set of permutation ๐ 75des ๐ descent number of permutation ๐ 76๐ท๐ lattice of divisors of ๐ 140๐ทโ divisibility poset on โ 181๐ท(๐) derangement number 43๐(๐) set of Dyck paths of semilength ๐ 26๐(๐) set of all digraphs on vertex set ๐ 21๐(๐, ๐) set of all digraphs on vertex set ๐ with ๐ edges 21deg๐ degree of a monomial 219deg ๐ฃ degree of vertex ๐ฃ in a graph 20ฮ๐(๐) forward difference operator of ๐(๐) 162๐ฟ๐ฅ,๐ฆ Kronecker delta 7๐ฟ(๐ฅ, ๐ง) delta function of poset incidence algebra 159๐ธ(๐บ) edge set of graph ๐บ 18๐ธ(๐ฟ) set structure on label set ๐ฟ 125๐ธ(๐ฟ) nonempty set structure on label set ๐ฟ 125๐ธ๐ Euler number 120๐๐ ๐th elementary symmetric function 221๐ธ(๐ก) generating function for elementary symmetric functions 221Exc๐ set of excedances of permutation ๐ 122exc ๐ number of excedances of permutation ๐ 122
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List of Notation xv
Symbol Definition Page
Fix ๐ fix point set of a function ๐ 44๐๐ Fibonacci number 3๐น๐ Fibonacci number 2๐ฝ๐ Galois field with ๐ elements 79๐(๐ฅ) ordinary generating function 81๐๐(๐ฅ) weight-generating function for weighted set ๐ 86๐น(๐) binomial poset ๐-interval factorial function 178๐น(๐ฅ) exponential generating function 117๐น๐ฎ(๐ฅ) exponential generating function for structure ๐ฎ 125๐น๐ fundamental quasisymmetric for set ๐ 269๐น๐ผ fundamental quasisymmetric for composition ๐ผ 269๐๐ number of standard Young tableaux of shape ๐ 225ฮฆ fundamental map on permutations 122๐ bijection between subsets and compositions 16๐บ โงต ๐ graph ๐บ with edge ๐ deleted 100๐บ/๐ graph ๐บ with edge ๐ contracted 101GL(๐) general linear group over vector space ๐ 287๐ข(๐) set of all graphs on vertex set ๐ 20๐ข(๐, ๐) set of all graphs on vertex set ๐ with ๐ edges 20๐บ๐ฅ stabilizer of element ๐ฅ under the action of group ๐บ 191๐ป๐ = ๐ป๐,๐ hook of cell ๐ = (๐, ๐) 230โ๐ = โ๐,๐ hooklength of cell ๐ = (๐, ๐) 230โ๐ set of hook diagrams with ๐ cells 278โ๐ ๐th complete homogeneous symmetric function 221๐ป(๐ก) complete homogeneous generating function 221ideg ๐ฃ in-degree of vertex ๐ฃ in a digraph 21Inv๐ inversion set of permutation ๐ 74inv ๐ inversion number of permutation ๐ 74โ(๐) incidence algebra of poset ๐ 158๐ผ(๐) lower-order ideal generated by ๐ in a poset 143ISF(๐บ; ๐ก) increasing spanning forest generating function of ๐บ 105ISF๐(๐บ) set of๐-edge increasing spanning forests of ๐บ 105isf๐(๐บ) number of๐-edge increasing spanning forests of ๐บ 105๐๐(๐บ) number of independent type ๐ partitions in graph ๐บ 254๐ฅ(๐) distributive lattice of lower-order ideals of poset ๐ 151๐พ๐ complete graph with ๐ vertices 19๐พ๐ lattice of compositions of ๐ 140๐พ๐,๐ number of tableaux of shape ๐ and content ๐ 225
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xvi List of Notation
Symbol Definition Page
๐ฟ(๐บ) Laplacian of graph ๐บ 62โ(๐บ) bond lattice of graph ๐บ 167โ(๐) set of linear extensions of ๐ 238โ(๐ถ) length of chain ๐ถ in a poset 147โ(๐) length of an integer partition ๐ 15โ(๐) length of a permutation ๐ 4lim๐โโ
๐๐(๐ฅ) limit of a sequence of formal power series 84lds ๐ length of a longest decreasing subsequence of ๐ 245lis ๐ length of a longest increasing subsequence of ๐ 244๐ฟ๐(๐) lattice of subspaces of ๐ฝ๐๐ 140๐ฟโ(๐) poset of subspaces of vector space ๐โ over ๐ฝ๐ 178๐ฟ(๐) lattice of subspaces of ๐ 140๐(๐น) type of partition induced by edge set ๐น 255๐! multiplicity factorial of partition ๐ 254maj ๐ major index of permutation ๐ 76๐(๐) Mertens function 183๐(๐) monomial quasisymmeric function for poset ๐ 275๐๐ผ monomial quasisymmetric function 268๐๐ monomial symmetric function 220๐(๐) Mรถbius function value on a poset ๐ 154๐(๐ฅ) one-variable Mรถbius function evaluated at ๐ฅ 154๐(๐ฅ, ๐ง) two-variable Mรถbius function on the interval [๐ฅ, ๐ง] 157โ nonnegative integers 1NBC๐(๐บ) set of no broken circuit sets of ๐ edges of ๐บ 102nbc๐(๐บ) number of no broken circuit sets of ๐ edges of ๐บ 102๐ฉโฐ(๐, ๐) set of ๐-๐ธ lattice paths from (0, 0) to (๐, ๐) 26odeg ๐ฃ out-degree of vertex ๐ฃ in a digraph 21๐ช๐ฅ orbit of an element ๐ฅ under action of a group 190๐(๐) big oh notation applied to function ๐ 182๐(๐) order of a group element ๐ 210โ positive integers 1๐โ dual of poset ๐ 142๐ซ๐ถ(๐บ) set of proper colorings of ๐บ with the positive integers 279๐(๐บ; ๐ก) chromatic polynomial of graph ๐บ 100Par ๐ set of ๐-partitions 238Par๐ ๐ set of ๐-partitions bounded by๐ 238๐๐ path with ๐ vertices 19๐(๐) set of partitions of the integer ๐ 13๐(๐) number of partitions of the integer ๐ 13๐๐ ๐th power sum symmetric function 221๐(๐ก) power sum symmetric generating function 221
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List of Notation xvii
Symbol Definition Page
๐(๐, ๐) set of partitions of ๐ into at most ๐ parts 15๐(๐, ๐) number of partitions of ๐ into at most ๐ parts 15๐(๐) permutations of a set ๐ 4๐(๐, ๐) permutations of length ๐ of a set ๐ 4๐((๐, ๐)) words of length ๐ over a set ๐ 5๐(๐) insertion tableau of ๐ 242๐ซ(๐ข; ๐ฃ) set of directed paths from ๐ข to ๐ฃ in a digraph 56ฮ ๐ partition lattice on [๐] 140ฮ (๐ฎ) partition structure on structure ๐ฎ 131ฮ ๐(๐ฎ) even partition structure on structure ๐ฎ 133ฮ ๐(๐ฎ) odd partition structure on structure ๐ฎ 133โ rational numbers 1๐(๐) set of compositions of the integer ๐ 16๐(๐) number of compositions of the integer ๐ 16๐(๐, ๐) set of compositions of ๐ into ๐ parts 16๐(๐, ๐) number of partitions of ๐ into ๐ parts 16QSym algebra of quasisymmetric functions 268QSym๐ quasisymmetric functions of degree ๐ 268๐(๐) recording tableau of ๐ 242๐๐(ฮ ) quasisymmetric function for patterns ฮ 277โ real numbers 1โ๐ถ(๐) set of functions reverse compatible with ๐ 270rk ๐ rank of a ranked poset ๐ 147Rk๐ ๐ ๐th rank set of a ranked poset ๐ 147rk ๐ฅ rank of an element ๐ฅ in a ranked poset 147โ(๐, ๐) set of partitions contained in a ๐ ร ๐ rectangle 79RPar ๐ set of reverse ๐-partitions 271โ(๐) reduced incidence algebra of a binomial poset 179rpp๐(๐) number of shape ๐ reverse plane partitions of ๐ 233rpar(๐; ๐ฑ) generating function for reverse ๐-partitions 271๐๐ฅ(๐) row insertion of element ๐ฅ into tableau ๐ 241๐(๐น) vertex partition induced by edge set ๐น 255๐โถ ๐บ โ GL(๐) representation of group ๐บ 287๐ฎ(๐ฟ) labeled structure on label set ๐ฟ 124๐ pattern poset 140๐๐ symmetric group on [๐] 11๐๐(๐) summation operator applied to function ๐(๐) 162sgn sign function on a signed set 44sh ๐ shape of tableau ๐ 225๐ (๐, ๐) signed Stirling number of the first kind 13
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xviii List of Notation
Symbol Definition Page
๐(๐, ๐) set of partitions of the set ๐ into ๐ blocks 10๐(๐, ๐) Stirling number of the second kind 10๐๐(๐ฟ, ๐) set of ordered partitions of the set ๐ฟ into ๐ blocks 127๐ฎ๐(๐บ) set of spanning trees of graph ๐บ 59st statistic on a set 74std ๐ standardization of the permutation ๐ 28Supp ๐ฅ support set of ๐ฅ in a product of claws 173supp ๐ฅ size of support set of ๐ฅ in a product of claws 173Sym algebra of symmetric functions 220Sym๐ symmetric functions of degree ๐ 220SYT(๐) set of standard Young tableaux of shape ๐ 224SSYT(๐) set of semistandard Young tableaux of shape ๐ 225๐ ๐ Schur function 225๐๐,๐ element in cell (๐, ๐) of tableau ๐ 225๐ฏ๐ set of monomino-domino tilings of a row of ๐ squares 3๐(๐) upper-order ideal generated by ๐ in a poset 143๐(๐ท) vertex set of digraph ๐ท 21๐(๐บ) vertex set of graph ๐บ 18๐โ vector space with a countably infinite basis over ๐ฝ๐ 178๐ค๐(๐) Whitney number of the first kind for a poset ๐ 156๐ ๐(๐) Whitney number of the second kind for a poset ๐ 156๐๐ walk with ๐ vertices 19wt weight function on a set 86๐ฑ a countably infinite set of variables 219๐ฑ๐ monomial for a coloring ๐ of a graph 253๐ฑ๐ monomial for a function ๐ 270๐ฑ๐ monomial for a tableau ๐ 225๐๐ fixed points of group element ๐ acting on set ๐ 192๐(๐บ; ๐ฑ) chromatic symmetric function of graph ๐บ 253๐(๐บ; ๐ฑ, ๐) chromatic quasisymmetric function of graph ๐บ 280๐ Youngโs lattice 140โค set of integers 1๐(๐ฅ, ๐ง) zeta function in the incidence algebra of a poset 159๐(๐ ) Riemann zeta function 182๐ง(๐) cycle index of group element ๐ 197๐(๐บ) cycle index of group ๐บ 197#๐ cardinality of the set ๐ 1|๐| size (sum of values) of a function 236|๐| cardinality of the set ๐ 1|๐| sum of entries of tableau ๐ 233๐ โ ๐ disjoint union of sets ๐ and ๐ 1
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List of Notation xix
Symbol Definition Page
|๐| sum of the parts of partition ๐ 13๐ โข ๐ ๐ is a partition of ๐ 13๐ ร ๐ (Cartesian) product of sets ๐ and ๐ 1๐ โ ๐ disjoint union of posets ๐ and ๐ 145๐ โ ๐ ordinal sum of posets ๐ and ๐ 146๐ ร ๐ (Cartesian) product of posets ๐ and ๐ 146[๐] linear transformation for group element ๐ 287[๐]๐ต matrix in basis ๐ต for group element ๐ 287[๐] set of integers {1, 2, . . . , ๐} 4[๐]๐ ๐-analogue of nonnegative integer ๐ 75[๐]๐! ๐-analogue of ๐! 75[๐ฅ๐]๐(๐ฅ) coefficient of ๐ฅ๐ in ๐(๐ฅ) 83๐โ๐ ๐ falling factorial with ๐ factors 42๐ set of subsets of ๐ 5(๐๐) set of ๐-element subsets of ๐ 6(๐๐) binomial coefficient 7[๐๐]๐ ๐-binomial coefficient 77
[๐๐] ๐-dimensional subspaces of vector space ๐ 79{{๐, ๐, . . . }} multiset individual element notation 8{{๐2, . . . }} multiset multiplicity notation 8((๐๐)) set of ๐-element multisubsets of ๐ 9๐(๐บ) chromatic number of ๐บ 99๐(๐) character of group element ๐ 291๐ฅ โ ๐ฆ ๐ฅ is covered by ๐ฆ in a poset 140๐ฆ โ ๐ฅ ๐ฆ covers ๐ฅ in a poset 1400ฬ the minimum element of a poset 1421ฬ the maximum element of a poset 142[๐ฅ, ๐ฆ] closed interval from ๐ฅ to ๐ฆ in a poset 143๐ฅ โง ๐ฆ meet of ๐ฅ and ๐ฆ in a poset 148โ๐ meet of the subset ๐ in a poset 149๐ฅ โจ ๐ฆ join of ๐ฅ and ๐ฆ in a poset 149๐ + ๐ sum of subspaces ๐ and ๐ 149๐ โ ๐ convolution of ๐ and ๐ in the incidence algebra 158๐(๐; ๐ก) characteristic polynomial of a ranked poset ๐ 164๐/ โผ quotient of poset ๐ by equivalence relation โผ 169๐๐ primitive ๐th root of unity 210๐ RSโฆ (๐,๐) RobinsonโSchensted map 242๐ RSKโฆ (๐,๐) RobinsonโSchenstedโKnuth map 244
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Chapter 1
Basic Counting
In this chapter we will develop the most elementary techniques for enumerating sets.Even though these methods are relatively basic, they will presage more complicatedthings to come. We denote the integers by โค and parameters such as ๐ and ๐ are alwaysassumed to be integral unless otherwise indicated. We also use the notation โ and โfor the nonnegative and positive integers, respectively. As usual, โ, โ, and โ standfor the rational numbers, real numbers, and complex numbers, respectively. Finally,whenever taking the cardinality of a set we will assume it is finite.
1.1. The Sum and Product Rules for sets
The SumandProduct Rules for sets are the basis formuch of enumeration. Andwewillsee various extensions of them later to ordinary and exponential generating functions.Although the rules are very easy to prove, we will include the demonstrations becausethe results are so useful. Given a finite set ๐, we will use either of the notations #๐ or|๐| for its cardinality. We will also write ๐ โ ๐ for the disjoint union of ๐ and ๐, andusage of this symbol implies disjointness even if it has not been previously explicitlystated. Finally, our notation for the (Cartesian) product of sets is
๐ ร ๐ = {(๐ , ๐ก) โฃ ๐ โ ๐, ๐ก โ ๐}.
Lemma 1.1.1. Let ๐, ๐ be finite sets.(a) (Sum Rule) If ๐ โฉ ๐ = โ , then
|๐ โ ๐| = |๐| + |๐|.(b) (Product Rule) For any finite sets
|๐ ร ๐| = |๐| โ |๐|.
Proof. Let ๐ = {๐ 1, . . . , ๐ ๐} and ๐ = {๐ก1, . . . , ๐ก๐}. For part (a), if ๐ and ๐ are disjoint,then we have ๐ โ ๐ = {๐ 1, . . . , ๐ ๐, ๐ก1, . . . , ๐ก๐} so that |๐ โ ๐| = ๐ + ๐ = |๐| + |๐|.
1
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2 1. Basic Counting
For part (b), we induct on ๐ = |๐|. If ๐ = โ , then ๐ ร ๐ = โ so that |๐ ร ๐| = 0 asdesired. If |๐| โฅ 1, then let ๐ โฒ = ๐ โ {๐ก๐}. We can write ๐ ร ๐ = (๐ ร ๐ โฒ) โ (๐ ร {๐ก๐}).Also ๐ ร {๐ก๐} = {(๐ 1, ๐ก๐), . . . , (๐ ๐, ๐ก๐)}, which has |๐| = ๐ elements since the secondcomponent is constant. Now by part (a) and induction
|๐ ร ๐| = |๐ ร ๐ โฒ| + |๐ ร {๐ก๐}| = ๐(๐ โ 1) + ๐ = ๐๐,
which finishes the proof. โก
In combinatorial choice problems, one is often given either the option to do oneoperation or another, or to do both. Suppose there are ๐ ways of doing the first oper-ation and ๐ ways of doing the second. If there is no common operation, then the SumRule tells us that the number of ways to do one or the other is๐+ ๐. And if doing thefirst operation has no effect on doing the second, then the Product Rule gives a countof ๐๐ for doing the first and then the second. More generally if there are ๐ ways ofdoing the first operation and, no matter which of the๐ is chosen, the number of waysto continue with the second operation is ๐, then again there are ๐๐ ways to do both.(The actual ๐ second operations availablemay depend on the choice of the first, but nottheir number.) So in practice one translates from English to mathematics by replacingโorโ with addition and โandโ with multiplication.
Another important concept related to cardinalities is that of a bijection. A bijectionbetween sets ๐, ๐ is a function ๐โถ ๐ โ ๐ which is both injective (one-to-one) and sur-jective (onto). If ๐, ๐ are finite, then the existence of a bijection between them impliesthat |๐| = |๐|. (One can extend this notion to infinite sets, but we will have no causeto do so here.) In combinatorics, one often uses bijections to prove that two sets havethe same cardinality. See, for just one of many examples, the proof of Theorem 1.1.2below.
We will illustrate these ideas with one of the most famous sequences in all of com-binatorics: the Fibonacci numbers. As is sometimes the case, there is an amusing (ifsomewhat improbable) story attached to the sequence. One starts at the beginning oftime with a pair of immature rabbits, one male and one female. It takes one monthfor rabbits to mature. In every subsequent month a pair gives birth to another pair ofimmature rabbits, one male and one female. If rabbits only breed with their birth part-ner and live forever (as I said, the story is somewhat improbable), how many pairs ofrabbits are there at the beginning ofmonth ๐? Let us call this number ๐น๐. It will be con-venient to let ๐น0 = 0. Since we begin with only one pair, ๐น1 = 1. And at the beginningof the second month, the pair has matured but produced no offspring, so ๐น2 = 1. Insubsequent months, one has all the rabbits from the previous month, counted by ๐น๐โ1,together with the newborn pairs. The number of newborn pairs equals the number ofmature pairs from the previous month, which equals the total number of pairs fromthe month before which is ๐น๐โ2. Thus, applying the Sum Rule,
(1.1) ๐น๐ = ๐น๐โ1 + ๐น๐โ2 for ๐ โฅ 2 with ๐น0 = 0 and ๐น1 = 1
where we can start the recursion at ๐ = 2 rather than ๐ = 3 due to letting ๐น0 = 0.The ๐น๐ are called the Fibonacci numbers. It is also important to note that some authors
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1.1. The Sum and Product Rules for sets 3
Figure 1.1. ๐ฏ3
define this sequence by letting
(1.2) ๐0 = ๐1 = 1 and ๐๐ = ๐๐โ1 + ๐๐โ2 for ๐ โฅ 2.So it is important to make sure which flavor of Fibonacci is being discussed in a givencontext.
One might wonder if there is an explicit formula for ๐น๐ in addition to the recursiveone above. We will see that such an expression exists, although it is far from obvioushow to derive it from what we have done so far. Indeed, we will need the theory ofordinary generating functions discussed in Chapter 3 to derive it.
Another thing which might be desired is a combinatorial interpretation for ๐น๐. Acombinatorial interpretation for a sequence of nonnegative integers ๐0, ๐1, ๐2, . . . is asequence of sets ๐0, ๐1, ๐2, . . . such that #๐๐ = ๐๐ for all ๐. Such interpretations of-ten give rise to very pretty and intuitive proofs about the original sequence and so arehighly desirable. One could argue that the story of the rabbits already gives such aninterpretation. But we would like something more amenable to mathematical manip-ulation.
Suppose we are given a row of squares. We are also given two types of tiles: domi-nos which can cover two squares and monominos which can cover one. A tiling of therow is a set of tiles which covers each square exactly once. Let ๐ฏ๐ be the set of tilingsof a row of ๐ squares. See Figure 1.1 for a list of the elements of ๐ฏ3. There is a simplerelationship between tilings and Fibonacci numbers.
Theorem 1.1.2. For ๐ โฅ 1 we have๐น๐ = #๐ฏ๐โ1.
Proof. It suffices to prove that both sides of this equation satisfy the same initial con-ditions and recurrence relation. When the row contains no squares, it only has theempty tiling so ๐ฏ0 = 1 = ๐น1. And when there is one square, it can only be tiled bya monomino so ๐ฏ1 = 1 = ๐น2. For the recursion, the tilings in ๐ฏ๐ can be divided intotwo types: those which end with a monomino and those which end with a domino.Removing the last tile shows that these tilings are in bijection with those in ๐ฏ๐โ1 andthose in ๐ฏ๐โ2, respectively. Thus #๐ฏ๐ = #๐ฏ๐โ1 + #๐ฏ๐โ2 as desired. โก
To see the power of a good combinatorial interpretation, we will now give a simpleproof of an identity for the ๐น๐. Such identities are legion. See, for example, the book ofBenjamin and Quinn [10].
Corollary 1.1.3. For๐ โฅ 1 and ๐ โฅ 0 we have๐น๐+๐ = ๐น๐โ1๐น๐ + ๐น๐๐น๐+1.
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4 1. Basic Counting
Proof. By the previous theorem, the left-hand side counts the number of tilings of arow of๐+๐โ1 squares. So it suffices to show that the same is true of the right. Labelthe squares 1, . . . , ๐ + ๐ โ 1 from left to right. We can write ๐ฏ๐+๐โ1 = ๐ฎ โ ๐ฏ where๐ฎ contains those tilings with a domino covering squares ๐ โ 1 and ๐, and ๐ฏ has thetilings with๐โ1 and๐ in different tiles. The tilings in๐ฏ are essentially pairs of tilings,the first covering the first๐โ 1 square and second covering the last ๐ squares. So theProduct Rule gives |๐ฏ| = |๐ฏ๐โ1| โ |๐ฏ๐| = ๐น๐๐น๐+1. Removing the given domino from thetilings in ๐ฎ again splits each tiling into a pair with the first covering๐โ2 squares andthe second ๐ โ 1. Taking cardinalities results in |๐ฎ| = ๐น๐โ1๐น๐. Finally, applying theSum Rule finishes the proof. โก
The demonstration just given is called a combinatorial proof since it involves count-ing discrete objects. We will meet other useful proof techniques as we go along. Butcombinatorial proofs are often considered to be themost pleasant, in part because theycan be more illuminating than demonstrations just involving formal manipulations.
1.2. Permutations and words
It is always importantwhen considering an enumeration problem todeterminewhetherthe objects being considered are ordered or not. In this section we will consider themost basic ordered structures, namely permutations and words.
If ๐ is a set with #๐ = ๐, then a permutation of ๐ is a sequence ๐ = ๐1 . . . ๐๐obtained by listing the elements of ๐ in some order. If ๐ is a permutation, we willalways use ๐๐ to denote the ๐th element of ๐ and similarly for other ordered structures.We let ๐(๐) denote the set of all permutations of ๐. For example,
๐({๐, ๐, ๐}) = {๐๐๐, ๐๐๐, ๐๐๐, ๐๐๐, ๐๐๐, ๐๐๐}.Clearly #๐(๐) only depends on #๐. So often we choose the canonical ๐-element set
[๐] = {1, 2, . . . , ๐}.We can also consider ๐-permutations of ๐ which are sequences ๐ = ๐1 . . . ๐๐ obtainedby linearly ordering ๐ distinct elements of ๐. Here, ๐ is called the length of the permu-tation and we write โ(๐) = ๐. Again, we use the same terminology and notation forother ordered structures. The set of all ๐-permutations of ๐ is denoted ๐(๐, ๐). By wayof illustration,
๐({๐, ๐, ๐, ๐}, 2) = {๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐, ๐๐}.In particular, if #๐ = ๐, then ๐(๐, ๐) = ๐(๐). Also ๐(๐, ๐) = โ for ๐ > ๐ since in thiscase it is impossible to pick ๐ distinct elements from a set with only ๐. And ๐(๐, 0) = {๐}where ๐ is the empty sequence.
To count permutations it will be convenient to introduce the following notation.Given nonnegative integers ๐, ๐, we can form the falling factorial
๐โ๐= ๐(๐ โ 1) . . . (๐ โ ๐ + 1).Note that ๐ equals the number of factors in the product.
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1.3. Combinations and subsets 5
Theorem 1.2.1. For ๐, ๐ โฅ 0 we have#๐([๐], ๐) = ๐โ๐ .
In particular#๐([๐]) = ๐! .
Proof. Since ๐([๐]) = ๐([๐], ๐), it suffices to prove the first formula. Given ๐ =๐1 . . . ๐๐ โ ๐([๐], ๐), there are ๐ ways to pick ๐1. Since ๐2 โ ๐1, there remains ๐ โ 1choices for ๐2. Since the number of choices for ๐2 does not depend on the actual ele-ment chosen for ๐, one can continue in this way and apply a modified version of theProduct Rule to obtain the result. โก
Note that when 0 โค ๐ โค ๐ we can write
(1.3) ๐โ๐=๐!
(๐ โ ๐)! .
But for ๐ > ๐ the product ๐ โ๐ still makes sense, even though the product cannot beexpressed as a quotient of factorials. Indeed, if ๐ > ๐, then zero is a factor and so๐ โ๐= 0, which agrees with the fact that ๐([๐], ๐) = โ . In the special case ๐ = 0 wehave ๐โ๐= 1 because it is an empty product. Again, this reflects the combinatorics inthat #๐([๐], 0) = {๐}.
One of the other things to keep track of in a combinatorial problem is whetherelements are allowed to be repeated or not. In permutations we have no repetitions.But the case when they are allowed is interesting as well. A ๐-word over a set ๐ is asequence ๐ค = ๐ค1 . . . ๐ค๐ where ๐ค๐ โ ๐ for all ๐. Note that there is no assumption thatthe ๐ค๐ are distinct. We denote the set of ๐-words over ๐ by ๐((๐, ๐)). Note the use ofthe double parentheses to denote the fact that repetitions are allowed. Note also that๐(๐, ๐) โ ๐((๐, ๐)), but usually the inclusion is strict. To illustrate
๐(({๐, ๐, ๐, ๐}, 2)) = ๐({๐, ๐, ๐, ๐}, 2) โ {๐๐, ๐๐, ๐๐, ๐๐}.The proof of the next result is almost identical to that of Theorem 1.2.1 and so is left tothe reader. When a result is given without proof, this is indicated by a box at the endof its statement.
Theorem 1.2.2. For ๐, ๐ โฅ 0 we have#๐(([๐], ๐)) = ๐๐. โก
1.3. Combinations and subsets
We will now consider unordered versions of the combinatorial objects studied in thelast section. These are sometimes called combinations, although the reader may knowthem by their more familiar name: subsets.
Given a set ๐, we let 2๐ denote the set of all subsets of ๐. Notice that 2๐ is a set, nota number. For example,
2{๐,๐,๐} = {โ , {๐}, {๐}, {๐}, {๐, ๐}, {๐, ๐}, {๐, ๐}, {๐, ๐, ๐}}.The reason for this notation should be made clear by the following result.
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6 1. Basic Counting
Theorem 1.3.1. For ๐ โฅ 0 we have#2[๐] = 2๐.
Proof. By Theorem 1.2.2 we have 2๐ = #๐(({0, 1}, ๐)). So it suffices to find a bijection๐โถ 2[๐] โ ๐(({0, 1}, ๐)),
and there is a canonical one. In particular, if ๐ โ [๐], then we let ๐(๐) = ๐ค1 . . . ๐ค๐where, for all ๐,
๐ค๐ = {1 if ๐ โ ๐,0 if ๐ โ ๐.
To show that ๐ is bijective, it suffices to find its inverse. If๐ค = ๐ค1 . . . ๐ค๐ โ ๐(({0, 1}, ๐)),then we let ๐โ1(๐ค) = ๐ where ๐ โ ๐ if ๐ค๐ = 1 and ๐ โ ๐ if ๐ค๐ = 0 where 1 โค ๐ โค ๐. It iseasy to check that the compositions ๐ โ ๐โ1 and ๐โ1 โ ๐ are the identity maps on theirrespective domains. This completes the proof. โก
The proof just given is called a bijective proof and it is a particularly nice kind ofcombinatorial proof. This is because bijective proofs can relate different types of com-binatorial objects, sometimes revealing unexpected connections. Also note that weproved ๐ bijective by finding its inverse rather than showing directly that it was one-to-one and onto. This is the preferred method as having a concrete description of ๐โ1can be useful later. Finally, when dealing with functions we will always compose themright-to-left so that
(๐ โ ๐)(๐ฅ) = ๐(๐(๐ฅ)).We now want to count subsets by their cardinality. For a set ๐ we will use the
notation
(๐๐) = {๐ โ ๐ โฃ #๐ = ๐}.
As an example,
({๐, ๐, ๐}2 ) = {{๐, ๐}, {๐, ๐}, {๐, ๐}}.
As expected, we now find the cardinality of this set.
Theorem 1.3.2. For ๐, ๐ โฅ 0 we have
#([๐]๐ ) =๐โ๐๐! .
Proof. Cross-multiplying and using Theorem 1.2.1 we see that it suffices to prove
#๐([๐], ๐) = ๐! โ #([๐]๐ ).
To see this, note that we can get each ๐1 . . . ๐๐ โ ๐([๐], ๐) exactly once by runningthrough the subsets ๐ = {๐ 1, . . . , ๐ ๐} โ [๐] and then ordering each ๐ in all possibleways. The number of choices for ๐ is #([๐]๐ ) and, by Theorem 1.2.1 again, the numberof ways of permuting the elements of ๐ is ๐!. So we are done by the Product Rule. โก
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1.3. Combinations and subsets 7
11 1
1 2 11 3 3 1
1 4 6 4 1
Figure 1.2. Rows 0 through 4 of Pascalโs triangle
Given ๐, ๐ โฅ 0, we define the binomial coefficient
(1.4) (๐๐) = #([๐]๐ ) =
๐โ๐๐! .
The reason for this name is that these numbers appear in the binomial expansionwhichwill be studied in Chapter 3. Often you will see the binomial coefficients displayed ina triangular array called Pascalโs trianglewhich has (๐๐) as the entry in the ๐th row and๐th diagonal. When ๐ > ๐ it is traditional to omit the zeros. See Figure 1.2 for rows 0through 4. (We apologize to the reader for not writing out the whole triangle, but thispage is not big enough.) For 0 โค ๐ โค ๐ we can use (1.3) to write
(1.5) (๐๐) =๐!
๐! (๐ โ ๐)! ,
which is pleasing because of its symmetry. We can also extend the binomial coefficientsto ๐ < 0 by letting (๐๐) = 0. This is in keeping with the fact that (
[๐]๐ ) = โ in this case.
In the next theorem, we collect various basic results about binomial coefficientswhichwill be useful in the sequel. In it, wewill use theKronecker delta function definedby
๐ฟ๐ฅ,๐ฆ = {1 if ๐ฅ = ๐ฆ,0 if ๐ฅ โ ๐ฆ.
Also note that we do not specify the range of the summation variable ๐ in (c) and (d)because it can be taken as either 0 โค ๐ โค ๐ or ๐ โ โค since the extra terms in the largersum are all zero. Both viewpoints will be useful on occasion.
Theorem 1.3.3. Suppose ๐ โฅ 0.(a) The binomial coefficients satisfy the initial condition
(0๐) = ๐ฟ๐,0
and recurrence relation
(๐๐) = (๐ โ 1๐ โ 1) + (
๐ โ 1๐ )
for ๐ โฅ 1.(b) The binomial coefficients are symmetric, meaning that
(๐๐) = (๐
๐ โ ๐).
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8 1. Basic Counting
(c) We have
โ๐(๐๐) = 2
๐.
(d) We have
โ๐(โ1)๐(๐๐) = ๐ฟ๐,0.
Proof. (a) The initial condition is clear. For the recursion let ๐ฎ1 be the set of ๐ โ ([๐]๐ )with ๐ โ ๐, and let ๐ฎ2 be the set of ๐ โ ([๐]๐ ) with ๐ โ ๐. Then (
[๐]๐ ) = ๐ฎ1 โ ๐ฎ2. But
if ๐ โ ๐, then ๐ โ {๐} โ ([๐โ1]๐โ1 ). This gives a bijection between ๐ฎ1 and ([๐โ1]๐โ1 ) so that
#๐ฎ1 = (๐โ1๐โ1). On the other hand, if๐ โ ๐, then ๐ โ ([๐โ1]๐ ) and this implies#๐ฎ2 = (
๐โ1๐ ).
Applying the Sum Rule completes the proof.
(b) It suffices to find a bijection ๐โถ ([๐]๐ ) โ ([๐]๐โ๐). Consider themap ๐โถ 2
[๐] โ 2[๐]by ๐(๐) = [๐] โ ๐ where the minus sign indicates difference of sets. Note that thecomposition ๐2 is the identity map so that ๐ is a bijection. Furthermore ๐ โ ([๐]๐ ) if andonly if ๐(๐) โ ( [๐]๐โ๐). So ๐ restricts to a bijection between these two sets.
(c) This follows by applying the Sum Rule to the equation 2[๐] = โจ๐ ([๐]๐ ).
(d) The case ๐ = 0 is easy, so we assume ๐ > 0. We will learn general techniquesfor dealing with equations involving signs in the next chapter. But for now, we try toprove the equivalent equality
โ๐ odd
(๐๐) = โ๐ even(๐๐).
Let๐ฏ1 be the set of ๐ โ 2[๐] with#๐ odd and let๐ฏ2 be the set of ๐ โ 2[๐] with#๐ even.We wish to find a bijection ๐โถ ๐ฏ1 โ ๐ฏ2. Consider the operation of symmetric difference
๐ ฮ ๐ = (๐ โ ๐) โ (๐ โ ๐).It is not hard to see that (๐ ฮ ๐) ฮ ๐ = ๐. Now define ๐โถ 2[๐] โ 2[๐] by ๐(๐) = ๐ ฮ {๐}so that, by the previous sentence, ๐2 is the identity. Furthermore, ๐ reverses parity andso restricts to the desired bijection. โก
As with the case of permutations and words, we want to enumerate โsetsโ whererepetitions are allowed. A multiset ๐ is an unordered collection of elements whichmay be repeated. For example
๐ = {{๐, ๐, ๐, ๐, ๐, ๐}} = {{๐, ๐, ๐, ๐, ๐, ๐}}.Note the use of double curly brackets to denote amultiset. Wewill also usemultiplicitynotation where ๐๐ denotes๐ copies of the element ๐. Continuing our example
๐ = {{๐3, ๐, ๐2}}.As with powers, an exponent of one is optional and an exponent of zero indicates thatthere are no copies of that element in the multiset. The cardinality of a multiset is itsnumber of elements counted with multiplicity. So in our example #๐ = 2+1+3 = 6.
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1.3. Combinations and subsets 9
If ๐ is a set, then๐ is a multiset on ๐ if every element of๐ is an element of ๐. We let((๐๐)) be the set of all multisets on ๐ of cardinality ๐ and
((๐๐)) = #(([๐]๐ )).
To illustrate
(({๐, ๐, ๐}2 )) = { {{๐, ๐}}, {{๐, ๐}}, {{๐, ๐}}, {{๐, ๐}}, {{๐, ๐}}, {{๐, ๐}} }
and so ((32)) = 6.
Theorem 1.3.4. For ๐, ๐ โฅ 0 we have
((๐๐)) = (๐ + ๐ โ 1
๐ ).
Proof. We wish to find a bijection
๐โถ (([๐]๐ )) โ ([๐ + ๐ โ 1]
๐ ).
Given a multiset๐ = {{๐1 โค ๐2 โค ๐3 โค โฏ โค ๐๐}} on [๐], let๐(๐) = {๐1 < ๐2 + 1 < ๐3 + 2 < โฏ < ๐๐ + ๐ โ 1}.
Now the๐๐+๐ โ 1 are distinct, and the fact that๐๐ โค ๐ implies๐๐+๐โ1 โค ๐+๐โ1.It follows that ๐(๐) โ ([๐+๐โ1]๐ ) and so the map is well-defined. It should now be easyfor the reader to construct an inverse, proving that ๐ is bijective. โก
As with the binomial coefficients, we extend ((๐๐)) to negative ๐ by letting it equalzero. In the future we will do the same for other constants whose natural domain ofdefinition is ๐, ๐ โฅ 0 without comment.
We do wish to comment on an interesting relationship between counting sets andmultisets. Note that definition (1.4) is well-defined for any complex number ๐ since thefalling factorial is just a product, and in particular it makes sense for negative integers.In fact, if ๐ โ โ, then
(โ๐๐ ) =(โ๐)(โ๐ โ 1)โฏ (โ๐ โ ๐ + 1)
๐!(1.6)
= (โ1)๐ ๐(๐ + 1)โฏ (๐ + ๐ โ 1)๐!
= (โ1)๐((๐๐))
by Theorem 1.3.4. This kind of situation where evaluation of an enumerative formulaat negative arguments yields, up to sign, another enumerative function is called com-binatorial reciprocity and will be studied in Section 3.9.
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10 1. Basic Counting
1.4. Set partitions
We have already seen that disjoint unions are nice combinatorially. So it should comeas no surprise that set partitions also play an important role.
A partition of a set ๐ is a set ๐ of nonempty subsets ๐ต1, . . . , ๐ต๐ such that ๐ = โจ๐ ๐ต๐,written ๐ โข ๐. The ๐ต๐ are called blocks and we use the notation ๐ = ๐ต1/ . . . /๐ต๐leaving out all curly brackets and commas, even though the elements of the blocks,as well as the blocks themselves, are unordered. For example, one set partition of๐ = {๐, ๐, ๐, ๐, ๐, ๐, ๐} is
๐ = ๐๐๐/๐๐/๐/๐ = ๐/๐๐/๐/๐๐๐.We let ๐ต(๐) be the set of all ๐ โข ๐. To illustrate,
๐ต({๐, ๐, ๐}) = {๐/๐/๐, ๐๐/๐, ๐๐/๐, ๐/๐๐, ๐๐๐}.The ๐th Bell number is ๐ต(๐) = #๐ต([๐]). Although there is no known expression for๐ต(๐) as a simple product, there is a recursion.
Theorem 1.4.1. The Bell numbers satisfy the initial condition ๐ต(0) = 1 and the recur-rence relation
๐ต(๐) = โ๐(๐ โ 1๐ โ 1)๐ต(๐ โ ๐)
for ๐ โฅ 1.
Proof. The initial condition counts the empty partition of โ . For the recursion, given๐ โ ๐ต([๐]), let ๐ be the number of elements in the block ๐ต containing ๐. Then thereare (๐โ1๐โ1) ways to pick the remaining ๐ โ 1 elements of [๐ โ 1] to be in ๐ต. And thenumber of ways to partition [๐] โ ๐ต is ๐ต(๐ โ ๐). Summing over all possible ๐ finishesthe proof. โก
We may sometimes want to keep track of the number of blocks in our partitions.So define ๐(๐, ๐) to be the set of all ๐ โข ๐ with ๐ blocks. The Stirling numbers of thesecond kind are ๐(๐, ๐) = #๐([๐], ๐). We will introduce Stirling numbers of the firstkind in the next section. For example
๐({๐, ๐, ๐}, 2) = {๐๐/๐, ๐๐/๐, ๐/๐๐}so ๐(3, 2) = 3. Just as with the binomial coefficients, the ๐(๐, ๐) for 1 โค ๐ โค ๐ canbe displayed in a triangle as in Figure 1.3. And like the binomial coefficients, theseStirling numbers satisfy a simple recurrence relation.
11 1
1 3 11 7 6 1
1 15 25 10 1
Figure 1.3. Rows 1 through 5 of Stirlingโs second triangle
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1.5. Permutations by cycle structure 11
Theorem 1.4.2. The Stirling numbers of the second kind satisfy the initial condition๐(0, ๐) = ๐ฟ๐,0
and recurrence relation
๐(๐, ๐) = ๐(๐ โ 1, ๐ โ 1) + ๐๐(๐ โ 1, ๐)for ๐ โฅ 1.
Proof. By now, the reader should be able to explain the initial condition without diffi-culty. For the recursion, the elements ๐ โ ๐([๐], ๐) are of two flavors: those where ๐ isin a block by itself and those where ๐ is in a block with other elements. Removing ๐ inthe first case leaves a partition in ๐([๐โ 1], ๐ โ 1) and this is a bijection. This accountsfor the summand ๐(๐โ1, ๐โ1). Removing ๐ in the second case leaves ๐ โ ๐([๐โ1], ๐),but this map is not a bijection. In particular, given ๐, one can insert ๐ into any one ofits ๐ blocks to recover an element of ๐([๐], ๐). So the total count is ๐๐(๐ โ 1, ๐) for thiscase. โก
1.5. Permutations by cycle structure
The ordered analogue of a decomposition of a set into a partition is the decompositionof a permutation of [๐] into cycles. These are counted by the Stirling numbers of thefirst kind.
The symmetric group is๐๐ = ๐([๐]). As the name implies,๐๐ has a group structuredefined as follows. If ๐ = ๐1 . . . ๐๐ โ ๐๐, then we can view this permutation as abijection ๐โถ [๐] โ [๐] where ๐(๐) = ๐๐. From this it follows that ๐๐ is a group wherethe operation is composition of functions.
Given ๐ โ ๐๐ and ๐ โ [๐], there is a smallest exponent โ โฅ 1 such that ๐โ(๐) = ๐.This and various other claims below will be proved using digraphs in Section 1.9. Inthis case, the elements ๐, ๐(๐), ๐2(๐), . . . , ๐โโ1(๐) are all distinct and we write
๐ = (๐, ๐(๐), ๐2(๐), . . . , ๐โโ1(๐))and call this a cycle of length โ or simply an โ-cycle of ๐. Cycles of length one are calledfixed points. As an example, if ๐ = 6514237 and ๐ = 1, then we have ๐(1) = 6, ๐2(1) =3, ๐3(1) = 1 so that ๐ = (1, 6, 3) is a cycle of ๐. We now iterate this process: if thereis some ๐ โ [๐] which is not in any of the cycles computed so far, we find the cyclecontaining ๐ and continue until every element is in a cycle. The cycle decompositionof ๐ is ๐ = ๐1 . . . ๐๐ where the ๐๐ are the cycles found in this process. Continuing ourexample, we could get
๐ = (1, 6, 3)(2, 5)(4)(7).To distinguish the cycle decomposition of๐ from its description as๐ = ๐1 . . . ๐๐wewillcall the latter the one-line notation for ๐. This is also distinct from two-line notation,which is where one writes
(1.7) ๐ = 1 2 . . . ๐๐1 ๐2 . . . ๐๐.
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12 1. Basic Counting
11 1
2 3 16 11 6 1
24 50 35 10 1
Figure 1.4. Rows 1 through 5 of Stirlingโs first triangle
Note that an โ-cycle can be written in โ different ways depending on which of itselements one starts with; for example
(1, 6, 3) = (6, 3, 1) = (3, 1, 6).Furthermore, the distinct cycles of ๐ are disjoint. So if we think of the cycle ๐ as thepermutation of [๐]which agrees with ๐ on the elements of ๐ and has all other elementsas fixed points, then the cycles of๐ = ๐1 . . . ๐๐ commute where we consider the productas a composition of permutations. Returning to our running example, we could write
๐ = (1, 6, 3)(2, 5)(4)(7) = (4)(1, 6, 3)(7)(2, 5) = (5, 2)(3, 1, 6)(7)(4).As mentioned above, we defer the proof of the following result until Section 1.9.
Theorem 1.5.1. Every ๐ โ ๐๐ has a cycle decomposition ๐ = ๐1 . . . ๐๐ which is uniqueup to the order of the factors and cyclic reordering of the elements within each ๐๐.
We are now in a position to proceed parallel to the development of set partitionswith a given number of blocks in the previous section. For ๐ โฅ 0we denote by ๐([๐], ๐)the set of all permutations in๐๐ which have ๐ cycles in their decomposition. Note thedifference between โ๐ cyclesโ referring to the number of cycles and โ๐-cyclesโ referringto the length of the cycles. The signless Stirling numbers of the first kind are ๐(๐, ๐) =#๐([๐], ๐). So, analogous to what we have seen before, ๐(๐, ๐) = 0 for ๐ < 0 or ๐ > ๐.To illustrate the notation,
๐([4], 1) = {(1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2), (1, 4, 2, 3), (1, 4, 3, 2)}so ๐(4, 1) = 6. In general, as youwill be asked to prove in an exercise, ๐([๐], 1) = (๐โ1)!.Part of Stirlingโs first triangle is displayed in Figure 1.4. We also have a recursion.
Theorem1.5.2. The signless Stirling numbers of the first kind satisfy the initial condition๐(0, ๐) = ๐ฟ๐,0
and recurrence relation๐(๐, ๐) = ๐(๐ โ 1, ๐ โ 1) + (๐ โ 1)๐(๐ โ 1, ๐)
for ๐ โฅ 1.
Proof. As usual, we concentrate on the recurrence. Given ๐ โ ๐([๐], ๐), we can re-move ๐ from its cycle. If ๐ was a fixed point, then the resulting permutations arecounted by ๐(๐ โ 1, ๐ โ 1). If ๐ was in a cycle of length at least two, then the per-mutations obtained upon removal are in ๐([๐ โ 1], ๐). So one must find the number of
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1.6. Integer partitions 13
ways to insert ๐ into a cycle of some ๐ โ ๐([๐โ1], ๐). There are โ places to insert ๐ in acycle of length โ. So the total number of insertion spots is the sum of the cycle lengthsof ๐, which is ๐ โ 1. โก
The reader may have guessed that there are also (signed) Stirling numbers of thefirst kind defined by
๐ (๐, ๐) = (โ1)๐โ๐๐(๐, ๐).It is not immediately apparent why one would want to attach signs to these constants.We will see one reason in Chapter 5 where it will be shown that the ๐ (๐, ๐) are theWhitney numbers of the first kind for the lattice of set partitions ordered by refinement.Here we will content ourselves with proving an analogue of part (d) of Theorem 1.3.3.
Corollary 1.5.3. For ๐ โฅ 0 we have
โ๐๐ (๐, ๐) = { 1 if ๐ = 0 or 1,0 if ๐ โฅ 2.
Proof. The cases when ๐ = 0 or 1 are easy to verify, so assume ๐ โฅ 2. Since ๐ (๐, ๐) =(โ1)๐โ๐๐(๐, ๐) and (โ1)๐ is constant throughout the summation, it suffices to showthatโ๐(โ1)๐๐(๐, ๐) = 0. Using Theorem 1.5.2 and induction on ๐ we obtain
โ๐(โ1)๐๐(๐, ๐) = โ
๐(โ1)๐๐(๐ โ 1, ๐ โ 1) +โ
๐(โ1)๐(๐ โ 1)๐(๐ โ 1, ๐)
= โโ๐(โ1)๐โ1๐(๐ โ 1, ๐ โ 1) + (๐ โ 1)โ
๐(โ1)๐๐(๐ โ 1, ๐)
= โ0 + (๐ โ 1)0
= 0
as desired. โก
Note the usefulness of considering the sums in the preceding proof as over ๐ โ โคrather than 0 โค ๐ โค ๐. This does away with having to consider any special cases at thevalues ๐ = 0 or ๐ = ๐.
1.6. Integer partitions
Just as one can partition a set into blocks, one can partition a nonnegative integer asa sum. Integer partitions play an important role not just in combinatorics but also innumber theory and the representation theory of the symmetric group. See the appendixat the end of the book for more information on the latter.
An integer partition of ๐ โฅ 0 is a multiset ๐ of positive integers such that the sumof the elements of ๐, denoted |๐|, is ๐. We also write ๐ โข ๐. These elements are calledthe parts. Since the parts of ๐ are unordered, we will always list them in a canonicalorder ๐ = (๐1, . . . , ๐๐) which is weakly decreasing. We let ๐(๐) denote the set of all
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14 1. Basic Counting
partitions of ๐ and ๐(๐) = #๐(๐). For example,๐(4) = {(1, 1, 1, 1), (2, 1, 1), (2, 2), (3, 1), (4)}
so that ๐(4) = 5. Note the distinction between ๐([๐]), which is a set of set partitions,and ๐(๐), which is a set of integer partitions. Sometimes we will just say โpartitionโif the context makes it clear whether we are partitioning sets or integers. We will usemultiplicity notation for integer partitions just as we would for any multiset, writing
๐ = (1๐1 , 2๐2 , . . . , ๐๐๐)where๐๐ is the multiplicity of ๐ in ๐.
There is no known product formula for ๐(๐). In fact, there is not even a simplerecurrence relation. One can use generating functions to derive results about thesenumbers, but that must wait until Chapter 3. Here we will just introduce a usefulgeometric device for studying ๐(๐). The Ferrers or Young diagram of ๐ = (๐1, . . . , ๐๐) โข๐ is an array of ๐ boxes into left-justified rows such that row ๐ contains ๐๐ boxes. Dotsare also sometimes used in place of boxes and in this case some authors use โFerrersdiagramโ for the dot variant and โYoung diagramโ for the corresponding array of boxes.We often make no distinction between a partition and its Young diagram. The Youngdiagram of ๐ = (5, 5, 2, 1) is shown in Figure 1.5. We should warn the reader thatwe are writing our Young diagrams in English notation where the rows are numberedfrom 1 to ๐ from the top down as in a matrix. Some authors prefer French notationwhere the rows are numbered from bottom to top as in a Cartesian coordinate system.The conjugate or transpose of ๐ is the partition ๐๐ก whose Young diagram is obtained byreflecting the diagram of ๐ about its main diagonal. This is done in Figure 1.5, showingthat (5, 5, 2, 1)๐ก = (4, 3, 2, 2, 2). There is also another way to express the parts of theconjugate.
๐ = (5, 5, 2, 1) = = โข โข โข โข โขโข โข โข โข โขโข โขโข
๐๐ก =
Figure 1.5. A partition, its Young diagram, and its conjugate
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1.6. Integer partitions 15
Proposition 1.6.1. If ๐ = (๐1, . . . , ๐๐) is a partition and ๐๐ก = (๐๐ก1, . . . , ๐๐ก๐ ), then, for1 โค ๐ โค ๐,
๐๐ก๐ = #{๐ โฃ ๐๐ โฅ ๐}.
Proof. By definition, ๐๐ก๐ is the length of the ๐th column of ๐. But that column containsa box in row ๐ if and only if ๐๐ โฅ ๐. โก
The number of parts of a partition ๐ is called its length and is denoted โ(๐). Atthis point the reader is probably expecting a discussion of those partitions of ๐ withโ(๐) = ๐. As it turns out, it is a bit simpler to consider ๐(๐, ๐), the set of all partitions ๐of๐with โ(๐) โค ๐, and๐(๐, ๐) = #๐(๐, ๐). Note that the number of ๐ โข ๐with โ(๐) = ๐is just ๐(๐, ๐) โ ๐(๐, ๐ โ 1). So in some sense the two viewpoints are equivalent. But itwill be easier to state our results in terms of ๐(๐, ๐). Note also that
๐(๐, 0) โค ๐(๐, 1) โค โฏ โค ๐(๐, ๐) = ๐(๐, ๐ + 1) = โฏ = ๐(๐).Because of this behavior, it is best to display the ๐(๐, ๐) in a matrix, rather than a trian-gle, keeping in mind that the entries in the ๐th row eventually stabilize to an infiniterepetition of the constant ๐(๐). Part of this array will be found in Figure 1.6. We alsoassume that ๐(๐, ๐) = 0 if ๐ < 0 or ๐ < 0. Unlike ๐(๐), one can write down a simplerecurrence relation for ๐(๐, ๐).
Theorem 1.6.2. The ๐(๐, ๐) satisfy
๐(0, ๐) = { 0 if ๐ < 0,1 if ๐ โฅ 0and
๐(๐, ๐) = ๐(๐ โ ๐, ๐) + ๐(๐, ๐ โ 1)for ๐ โฅ 1
Proof. We skip directly to the recursion. Note that since conjugation is a bijection,๐(๐, ๐) also counts the partitions ๐ = (๐1, . . . , ๐๐) โข ๐ such that ๐1 โค ๐. It will beconvenient to use this interpretation of ๐(๐, ๐) for the proof. We have two possiblecases. If ๐1 = ๐, then ๐ = (๐2, . . . , ๐๐) โข ๐ โ ๐ and ๐2 โค ๐1 = ๐. So these partitions arecounted by ๐(๐ โ ๐, ๐). The other possibility is that ๐1 โค ๐ โ 1. And these ๐ are takencare of by the ๐(๐, ๐ โ 1) term. โก
0 1 2 3 4 50 1 1 1 1 1 11 0 1 1 1 1 12 0 1 2 2 2 23 0 1 2 3 3 34 0 1 3 4 5 5
Figure 1.6. The values ๐(๐, ๐) for 0 โค ๐ โค 4 and 0 โค ๐ โค 5
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16 1. Basic Counting
1.7. Compositions
Recall that integer partitions are really unordered even though we usually list them inweakly decreasing fashion. This raises the question about what happens if we consid-ered ways to write ๐ as a sum when the summands are ordered. This is the notion of acomposition.
A composition of ๐ is a sequence ๐ผ = [๐ผ1, . . . , ๐ผ๐] of positive integers called partssuch thatโ๐ ๐ผ๐ = ๐. We write ๐ผ โง ๐ and use square brackets to distinguish composi-tions from integer partitions. This causes a notational conflict between [๐] as a compo-sition of ๐ and as the integers from 1 to ๐, but the context should make it clear whichinterpretation is meant. Let ๐(๐) be the set of compositions of ๐ and ๐(๐) = #๐(๐).So the compositions of 4 are
๐(4) = {[1, 1, 1, 1], [2, 1, 1], [1, 2, 1], [1, 1, 2], [2, 2], [3, 1], [1, 3], [4]}.
So ๐(4) = 8, which is a power of 2. This, as your author is fond of saying, is not acoincidence.
Theorem 1.7.1. For ๐ โฅ 1 we have
๐(๐) = 2๐โ1.
Proof. There is a famous bijection ๐ โถ 2[๐โ1] โ ๐(๐), which we will use to provethis result. This map will be useful when working with quasisymmetric functions inChapter 8. Given ๐ = {๐ 1, . . . , ๐ ๐} โ [๐ โ 1] written in increasing order, we define
(1.8) ๐(๐) = [๐ 1 โ ๐ 0, ๐ 2 โ ๐ 1, . . . , ๐ ๐ โ ๐ ๐โ1, ๐ ๐+1 โ ๐ ๐]
where, by definition, ๐ 0 = 0 and ๐ ๐+1 = ๐. To show that ๐ is well-defined, suppose๐(๐) = [๐ผ1, . . . , ๐ผ๐+1]. Since ๐ is increasing, ๐ผ๐ = ๐ ๐ โ ๐ ๐โ1 is a positive integer. Fur-thermore
๐+1โ๐=1
๐ผ๐ =๐+1โ๐=1
(๐ ๐ โ ๐ ๐โ1) = ๐ ๐+1 โ ๐ 0 = ๐.
Thus ๐(๐) โ ๐(๐) as desired.To show that ๐ is bijective, we construct its inverse ๐โ1 โถ ๐(๐) โ 2[๐โ1]. Given
๐ผ = [๐ผ1, . . . , ๐ผ๐+1] โ ๐(๐), we let
๐โ1(๐ผ) = {๐ผ1, ๐ผ1 + ๐ผ2, ๐ผ1 + ๐ผ2 + ๐ผ3, . . . , ๐ผ1 + ๐ผ2 +โฏ+ ๐ผ๐}.
It should not be hard for the reader to prove that ๐โ1 is well-defined and the inverse of๐. โก
As usual, we wish to make a more refined count by restricting the number of con-stituents of the object under consideration. Let ๐(๐, ๐) be the set of all compositionsof ๐ with exactly ๐ parts and let ๐(๐, ๐) = #๐(๐, ๐). Since the ๐(๐, ๐) will turn out tobe previously studied constants, we will forgo the usual triangle. The result below fol-lows easily by restricting the function ๐ from the previous proof, so the demonstrationis omitted.
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1.8. The twelvefold way 17
Theorem 1.7.2. The composition numbers satisfy๐(0, ๐) = ๐ฟ๐,0
and
๐(๐, ๐) = (๐ โ 1๐ โ 1)
for ๐ โฅ 1. โก
1.8. The twelvefold way
Wenowhave all the tools in place to count certain functions. There are 12 types of suchfunctions and so this scheme is called the twelvefoldway, an ideawhichwas introducedin a series of lectures by Gian-Carlo Rota. The namewas suggested by Joel Spencer andshould not be confused with the twelvefold path of Buddhism!
We will consider three types of functions ๐โถ ๐ท โ ๐ , namely, arbitrary functions,injections, and surjections. We will also permit the domain ๐ท and range ๐ to be oftwo types each: either distinguishable, which means it is a set, or indistinguishable,which means it is a multiset consisting of a single element repeated some number oftimes. Thus the total number of types of functions under consideration is the productof the number of choices for ๐, ๐ท, and ๐ or 3 โ 2 โ 2 = 12. Of course, a function wherethe domain or range is a multiset is not really well-defined, even though the intuitivenotion should be clear. To be precise, when ๐ท is a multiset and ๐ is a set, suppose ๐ทโฒis a set with |๐ทโฒ| = |๐ท|. Then a function ๐โถ ๐ท โ ๐ is an equivalence class of functions๐โถ ๐ทโฒ โ ๐ where ๐ and ๐ are equivalent if #๐โ1(๐) = #๐โ1(๐) for all ๐ โ ๐ . Thereader can come up with the corresponding notions for the other cases if desired. Wewill assume throughout that |๐ท| = ๐ and |๐ | = ๐ are both nonnegative integers. Wewill collect the results in the chart in Table 1.1.
We first deal with the case where both ๐ท and ๐ are distinguishable. Without lossof generality, we can assume that ๐ท = [๐]. So a function ๐โถ ๐ท โ ๐ can be consideredas a word ๐ค = ๐(1)๐(2) . . . ๐(๐). Since there are ๐ choices for each ๐(๐), we have, byTheorem 1.2.2, that the number of such ๐ is #๐(([๐], ๐)) = ๐๐. If ๐ is injective, then๐ค becomes a permutation, giving the count #๐([๐], ๐) = ๐โ๐ from Theorem 1.2.1. Forsurjective functions, we need a new concept. If ๐ท is a set, then the kernel of a function๐โถ ๐ท โ ๐ is the partition ker ๐ of๐ทwhose blocks are the nonempty subsets of the form๐โ1(๐) for ๐ โ ๐ . For example, if ๐โถ {๐, ๐, ๐, ๐} โ {1, 2, 3} is given by ๐(๐) = ๐(๐) = 2,
Table 1.1. The twelvefold way
๐ท ๐ arbitrary ๐ injective ๐ surjective ๐
dist. dist. ๐๐ ๐โ๐ ๐! ๐(๐, ๐)indist. dist. (๐+๐โ1๐ ) (
๐๐) (
๐โ1๐โ1)
dist. indist. โ๐๐=0 ๐(๐, ๐) ๐ฟ(๐ โค ๐) ๐(๐, ๐)indist. indist. ๐(๐, ๐) ๐ฟ(๐ โค ๐) ๐(๐, ๐) โ ๐(๐, ๐ โ 1)
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18 1. Basic Counting
๐(๐) = 3, and ๐(๐) = 1, then ker ๐ = ๐๐/๐/๐. If ๐ is to be surjective, then the functioncan be specified by picking a partition of๐ท for ker ๐ and then picking a bijection ๐ fromthe blocks of ker ๐ into ๐ . Continuing our example, ๐ is completely determined by itskernel and the bijection ๐(๐๐) = 2, ๐(๐) = 3, and ๐(๐) = 1. The number of ways tochoose ker ๐ = ๐ต1/ . . . /๐ต๐ is ๐(๐, ๐) by definition. And, using the injective case with๐ = ๐, the number of bijections ๐โถ {๐ต1, . . . , ๐ต๐} โ ๐ is ๐โ๐= ๐!. So the total count is๐! ๐(๐, ๐).
Now suppose ๐ท is indistinguishable and ๐ is distinguishable where we assume๐ = [๐]. Then one can think of ๐โถ ๐ท โ ๐ as a multiset ๐ = {{1๐1 , . . . , ๐๐๐ }} on ๐ where ๐๐ = #๐โ1(๐). It follows that โ๐๐๐ = #๐ท = ๐. So, by Theorem 1.3.4, thenumber of all such ๐ is
((๐๐)) = (๐ + ๐ โ 1
๐ ).
If ๐ is to be injective, then we are picking an ๐-element subset of ๐ = [๐] giving a countof (๐๐). If ๐ is to be surjective, then๐๐ โฅ 1 for all ๐ so that [๐1, . . . , ๐๐] is a compositionof ๐. It follows from Theorem 1.7.2 that the number of functions is ๐(๐, ๐) = (๐โ1๐โ1).
To deal with the case when ๐ท = [๐] is distinguishable and ๐ is indistinguishable,we introduce a useful extension of the Kronecker delta. If ๐ is any statement, we let
(1.9) ๐ฟ(๐) = { 1 if ๐ is true,0 if ๐ is false.Returning to our counting, ๐ is completely determined by its kernel, which is a parti-tion of [๐]. If we are considering all ๐, then the kernel can have any number of blocksup to and including ๐. Summing the corresponding Stirling numbers gives the corre-sponding entry in Table 1.1. If ๐ is injective, then for such a function to exist we musthave ๐ โค ๐. And in that case there is only one possible kernel, namely the partitioninto singleton blocks. This count can be summarized as ๐ฟ(๐ โค ๐). For surjective ๐ weare partitioning [๐] into exactly ๐ blocks, giving ๐(๐, ๐) possibilities.
If๐ท and ๐ are both indistinguishable, then the nonzero numbers of the form๐๐ =#๐โ1(๐) for ๐ โ ๐ completely determine ๐. And these numbers form a partition of๐ = #๐ท into at most ๐ = #๐ parts. Recalling the notation of Section 1.6, the totalnumber of such ๐ is ๐(๐, ๐). The line of reasoning for injective functions follows thatof the previous paragraph with the same resulting answer. Finally, for surjectivity weneed exactly ๐ parts, which is counted by ๐(๐, ๐) โ ๐(๐, ๐ โ 1).
1.9. Graphs and digraphs
Graph theory is a substantial part of combinatorics. We will use directed graphs togive the postponed proof of the existence and uniqueness of the cycle decompositionof permutations in ๐๐.
A labeled graph ๐บ = (๐, ๐ธ) consists of a set ๐ of elements called vertices and a set๐ธ of elements called edges where an edge consists of an unordered pair of vertices. Wewill write ๐(๐บ) and ๐ธ(๐บ) for the vertex and edge set of ๐บ, respectively, if we wish toemphasize the graph involved. Geometrically, we think of the vertices as nodes andthe edges as line segments or curves joining them. Conventionally, in graph theory an
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1.9. Graphs and digraphs 19
๐ฃ ๐ค
๐ฅ๐ฆ
Figure 1.7. A graph ๐บ
edge connecting vertices ๐ฃ and ๐ค is written ๐ = ๐ฃ๐ค rather than ๐ = {๐ฃ, ๐ค}. In this casewe say that ๐ contains ๐ฃ and ๐ค, or that ๐ has endpoints ๐ฃ and ๐ค. We also say that ๐ฃ and๐ค are neighbors. For example, a drawing of the graph ๐บ with vertices ๐ = {๐ฃ, ๐ค, ๐ฅ, ๐ฆ}and edges ๐ธ = {๐ฃ๐ค, ๐ฃ๐ฅ, ๐ฃ๐ฆ, ๐ค๐ฅ, ๐ฅ๐ฆ} is displayed in Figure 1.7. If #๐ = 1, then there isonly one graph with vertex set ๐ and such a graph is called trivial.
Call graph ๐ป a subgraph of ๐บ, written ๐ป โ ๐บ, if ๐(๐ป) โ ๐(๐บ) and ๐ธ(๐ป) โ ๐ธ(๐บ).In this case we also say that ๐บ contains ๐ป. There are several types of subgraphs whichwill play an important role in what follows. A walk of length โ in ๐บ is a sequence ofvertices๐ โถ ๐ฃ0, ๐ฃ1, . . . , ๐ฃโ such that ๐ฃ๐โ1๐ฃ๐ โ ๐ธ for 1 โค ๐ โค โ. We say that the walk isfrom ๐ฃ0 to ๐ฃโ, or is a ๐ฃ0โ๐ฃโ walk, or that ๐ฃ0, ๐ฃโ are the endpoints of๐ . We call๐ a pathif all the vertices are distinct and we usually use letters like ๐ for paths. In particular,we will use๐๐ or ๐๐ to denote a walk or a path having ๐ vertices, respectively. In ourexample graph, ๐ โถ ๐ฆ, ๐ฃ, ๐ฅ, ๐ค is a path of length 3 from ๐ฆ to๐ค. Notice that length refersto the number of edges in the path, which is one less than the number of vertices. Acycle of length โ in ๐บ is a sequence of distinct vertices ๐ถ โถ ๐ฃ1, ๐ฃ2, . . . , ๐ฃโ such that wehave distinct edges ๐ฃ๐โ1๐ฃ๐ for 1 โค ๐ โค โ, and subscripts are taken modulo โ so that๐ฃ0 = ๐ฃโ. Returning to our running example, ๐ถ โถ ๐ฃ, ๐ฅ, ๐ฆ is a cycle in ๐บ of length 3. In acycle the length is both the number of vertices and the number of edges. The notation๐ถ๐ will be used for a cycle with ๐ vertices and we will call this an ๐-cycle. We alsodenote by ๐พ๐ the complete graphwhich consists of ๐ vertices and all possible (๐2) edgesbetween them. A copy of a complete graph in a graph ๐บ is often called a clique. Thereis a close relationship between some of the parts of a graphwhich we have just defined.
Lemma 1.9.1. Let ๐บ be a graph and let ๐ข, ๐ฃ โ ๐ .
(a) Any walk from ๐ข to ๐ฃ contains a path from ๐ข to ๐ฃ.(b) The union of any two different paths from ๐ข to ๐ฃ contains a cycle.
Proof. We will prove (a) and leave (b) as an exercise. Let๐ โถ ๐ฃ0, . . . , ๐ฃโ be the walk.We will induct on โ, the length of๐ . If โ = 0, then๐ is a path. So assume โ โฅ 1. If๐is a path, then we are done. If not, then some vertex of๐ is repeated, say ๐ฃ๐ = ๐ฃ๐ for๐ < ๐. Then we have a ๐ขโ๐ฃ walk๐ โฒ โถ ๐ฃ0, ๐ฃ1, . . . , ๐ฃ๐, ๐ฃ๐+1, ๐ฃ๐+2, . . . , ๐ฃโ which is shorterthan๐ . By induction,๐ โฒ contains a path ๐ and so๐ contains ๐ as well. โก
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20 1. Basic Counting
To state our first graphical enumeration result, let ๐ข(๐) be the set of all graphs onthe vertex set ๐ . We will also use ๐ข(๐, ๐) to denote the set of all graphs in ๐ข(๐) with ๐edges.
Theorem 1.9.2. For ๐ โฅ 1 and ๐ โฅ 0 we have
#๐ข([๐]) = 2(๐2)
and
#๐ข([๐], ๐) = ((๐2)๐ ).
Proof. If ๐ = [๐] is given, then a graph ๐บ with vertex set ๐ is completely determinedby its edge set. Since there are ๐ vertices, there are (๐2) possible edges to choose from.So the number of ๐บ in ๐ข([๐]) is the number of subsets of these edges, which, by The-orem 1.3.1, is the given power of 2. The proof for ๐ข([๐], ๐) is similar, just using thedefinition (1.4). โก
Agraph is unlabeled if the vertices in๐ are indistinguishable. If the type of graph isclear from the context or does not matter for the particular application at hand, we willomit the adjectives โlabeledโ and โunlabeledโ. The enumeration of unlabeled graphsis much more complicated than for labeled ones. So this discussion is postponed untilSection 6.4 where we will develop the necessary tools.
If ๐บ is a graph and ๐ฃ โ ๐ , then the degree of ๐ฃ is
deg ๐ฃ = the number of ๐ โ ๐ธ containing ๐ฃ.
In our running example deg ๐ฃ = deg ๐ฅ = 3 and deg๐ค = deg ๐ฆ = 2. There is a nicerelationship between vertex degrees and the cardinality of the edge set. The demon-stration of the next result illustrates an important method of proof in combinatorics,counting in pairs.
Theorem 1.9.3. For any graph ๐บ we have
โ๐ฃโ๐
deg ๐ฃ = 2|๐ธ|.
Proof. Consider๐ = {(๐ฃ, ๐) | ๐ฃ is contained in ๐}.
Then#๐ = โ
๐ฃโ๐(number of ๐ containing ๐ฃ) = โ
๐ฃโ๐deg ๐ฃ.
On the other hand
#๐ = โ๐โ๐ธ
(number of ๐ฃ contained in ๐) = โ๐โ๐ธ
2 = 2|๐ธ|.
Equating the two counts finishes the proof. โก
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1.9. Graphs and digraphs 21
๐ฃ ๐ค
๐ฅ๐ฆ
Figure 1.8. A digraph ๐ท
Theorem 1.9.3 is often called the Handshaking Lemma because of the followinginterpretation. Suppose ๐ is the set of people at a party and we draw an edge betweenperson ๐ฃ and person ๐ค if they shake hands during the festivities. Then adding up thenumber of handshakes given by each person gives twice the total number of hand-shakes.
It is often useful to have specified directions along the edges. A labeled directedgraph, also called a digraph, is ๐ท = (๐, ๐ด) where ๐ is a set of vertices and ๐ด is a setof arcs which are ordered pairs of vertices. We use the notation ๐ = ๐ฃ๐ค for arcs andsay that ๐ goes from ๐ฃ to ๐ค. To illustrate, the digraph with ๐ = {๐ฃ, ๐ค, ๐ฅ, ๐ฆ} and ๐ด ={๐ฃ๐ค,๐ค๐ฃ,๐ค๐ฅ, ๐ฆ๐ฃ, ๐ฆ๐ฅ} is drawn in Figure 1.8. We use ๐(๐ท) and๐ด(๐ท) to denote the vertexset and arc set, respectively, of a digraph ๐ท when we wish to be more precise. Directedwalks, paths, and cycles are defined for digraphs similarly to their undirected cousins ingraphs, just insisting the ๐ฃ๐โ1๐ฃ๐ โ ๐ด for ๐ in the appropriate range. So, in our exampledigraph, ๐ โถ ๐ฆ, ๐ฃ, ๐ค, ๐ฅ is a directed path and ๐ถ โถ ๐ฃ,๐ค is a directed cycle. Note that๐ค, ๐ฅ, ๐ฆ, ๐ฃ is not a directed path because the arc between ๐ฅ and ๐ฆ goes the wrong way.
Let ๐(๐) and ๐(๐, ๐) be the set of digraphs and the set of digraphs with ๐ arcs,respectively, having vertex set ๐ . The next result is proved in much the same manneras Theorem 1.9.2 so the demonstration is omitted.
Theorem 1.9.4. For ๐ โฅ 1 and ๐ โฅ 0 we have#๐([๐]) = 2๐(๐โ1)
and
#๐([๐], ๐) = (๐(๐ โ 1)๐ ). โก
In a digraph ๐ท there are two types of degrees. Vertex ๐ฃ โ ๐ has out-degree andin-degree
odeg ๐ฃ = the number of ๐ โ ๐ด of the form ๐ = ๐ฃ๐ค,ideg ๐ฃ = the number of ๐ โ ๐ด of the form ๐ = ๐ค๐ฃ,
respectively. In Figure 1.8, for example, odeg ๐ฃ = 1 and ideg ๐ฃ = 2. The next result willpermit us to finish our leftover business from Section 1.5. The union of digraphs ๐ทโช๐ธis the digraph with vertices ๐(๐ทโช๐ธ) = ๐(๐ท)โช๐(๐ธ) and arcs ๐ด(๐ทโช๐ธ) = ๐ด(๐ท)โช๐ด(๐ธ).
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22 1. Basic Counting
Lemma 1.9.5. Let ๐ท = (๐, ๐ด) be a digraph. We have odeg ๐ฃ = ideg ๐ฃ = 1 for all ๐ฃ โ ๐if and only if ๐ท is a disjoint union of directed cycles.
Proof. The reverse implication is easy to see since the out-degree and in-degree of anyvertex ๐ฃ of ๐ท would be the same as those degrees in the directed cycle containing ๐ฃ.But in such a cycle odeg ๐ฃ = ideg ๐ฃ = 1.
For the forward direction, pick any ๐ฃ = ๐ฃ1 โ ๐ . Since odeg ๐ฃ1 = 1 there mustexist a vertex ๐ฃ2 with ๐ฃ1๐ฃ2 โ ๐ด. By the same token, there must be a ๐ฃ3 with ๐ฃ2๐ฃ3 โ ๐ด.Continue to generate a sequence ๐ฃ1, ๐ฃ2, . . . in this manner. Since ๐ is finite, there mustbe two indices ๐ < ๐ such that ๐ฃ๐ = ๐ฃ๐ . Let ๐ be the smallest such index and let ๐ bethe first index after ๐ where repetition occurs. Thus ๐ = 1, for if not, then we have๐ฃ๐โ1๐ฃ๐, ๐ฃ๐โ1๐ฃ๐ โ ๐ด, contradicting the fact that ideg ๐ฃ๐ = 1. By definition of ๐, we havea directed cycle ๐ถ โถ ๐ฃ1, ๐ฃ2, . . . , ๐ฃ๐โ1. Furthermore, no vertex of ๐ถ can be involved inanother arc since that would make its out-degree or in-degree too large. Continuing inthis manner, we can decompose ๐ท into disjoint directed cycles. โก
Sometimes it is useful to allow loops in a graph which are edges of the form ๐ = ๐ฃ๐ฃ.Similarly, we canpermit loops as arcs๐ = ๐ฃ๐ฃ in a digraph. Another possibility is thatwewould wantmultiple edges, meaning that one could have more than one edge betweena given pair of vertices, making ๐ธ into a multiset. Multiple arcs are defined similarly.If we make no specification for our (di)graph, then we are assuming that it has neitherloops nor multiple edges. We will now prove Theorem 1.5.1.
Proof (of Theorem 1.5.1). To any ๐ โ ๐๐ we associate its functional digraph ๐ท๐which has ๐ = [๐] and an arc โ๐ค๐ฅ โ ๐ด if and only if ๐(๐) = ๐. Now ๐ท๐ is a digraphwith loops. Because ๐ is a function we have odeg ๐ = 1 for all ๐ โ [๐]. And because ๐is a bijection we also have ideg ๐ = 1 for all ๐. The proof of the previous lemma worksequally well if one allows loops. So ๐ท๐ is a disjoint union of cycles. But cycles of thedigraph ๐ท๐ corre