Combinatorial Theory (Grundlehren der mathematischen Wissenschaften, 234) 9781461566687, 9781461566663, 1461566681

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Grundlehren der mathematischen Wissenschaften 234 A Series of Comprehensive Studies in Mathematics

Editors

S.S. Chern 1.L. Doob 1. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M.M. Postnikov W. Schmidt D.S. Scott K. Stein 1. Tits B.L. van der Waerden Managing Editors

B. Eckmann 1.K. Moser

Martin Aigner

Combinatorial Theory

Springer-Verlag Berlin Heidelberg New York

Martin Aigner II. Institut fiir Mathematik Freie Universitat Berlin Konigin-Luise-Strasse 24/26 1000 Berlin 33 Federal Republic of Germany

AMS Subject Classification (1980): 05xx, 06xx

With 123 Figures

Library of Congress Cataloging in Publication Data Aigner, Martin, 1942Combinatorial theory. (Grundlehren der mathematischen Wissenschaften; 234) Bibliography: p. Includes index. \. Combinatorial analysis. I. Title. II. Series: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; 234) QA164.A36 511'.6 79-1011

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.

© 1979 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1979 9 8 7 6 543 2 1

ISBN 978-1-4615-6668-7 DOl 10.1007/978-1-4615-6666-3

ISBN 978-1-4615-6666-3 (eBook)

Preface

It is now generally recognized that the field of combinatorics has, over the past years, evolved into a fully-fledged branch of discrete mathematics whose potential with respect to computers and the natural sciences is only beginning to be realized. Still, two points seem to bother most authors: The apparent difficulty in defining the scope of combinatorics and the fact that combinatorics seems to consist of a vast variety of more or less unrelated methods and results. As to the scope of the field, there appears to be a growing consensus that combinatorics should be divided into three large parts:

(a) Enumeration, including generating functions, inversion, and calculus of finite differences; (b) Order Theory, including finite posets and lattices, matroids, and existence results such as Hall's and Ramsey's; (c) Configurations, including designs, permutation groups, and coding theory. The present book covers most aspects of parts (a) and (b), but none of (c). The reasons for excluding (c) were twofold. First, there exist several older books on the subject, such as Ryser [1] (which I still think is the most seductive introduction to combinatorics), Hall [2], and more recent ones such as Cameron-Van Lint [1] on groups and designs, and Blake-Mullin [1] on coding theory, whereas no comprehensive book exists on (a) and (b). Second, the vast diversity of types of designs, the very complicated methods usually still needed to prove existence or nonexistence, and, in general, the rapid change this subject is presently undergoing do not favor a thorough treatment at this moment. I have also omitted reference to algorithms of any kind because I feel that presently nothing more can be said in book form about this subject beyond Knuth [1], Lawler [1], and Nijenhuis-Wilf [1]. As to the second point, that of systematizing the definitions, methods, and results into something resembling a theory, the present book tries to accomplish just this, admittedly at the expense of some of the spontaneity and ingenuity that makes combinatorics so appealing to mathematicians and non-mathematicians alike. To start with, mappings are grouped together into classes by placing various restrictions on them. To stick to the division outlined above, these classes of mappings are then counted, ordered, and arranged. The emphasis on ordering is well justified by the everyday experience of a combinatorist that most discrete structures, while perhaps lacking a simple algebraic structure, invariably admit

vi

Preface

a natural ordering. Following this program, the book is divided into three parts, the first part presenting the basic material on mappings and posets, in Chapters I and II, respectively, the second part dealing with enumeration in Chapters III to V, and the third part on the order-theoretical aspects in Chapters VI-VIII. The arrangement of the material allows the reader to use the three parts almost independently and to combine several subsections into a course on special topics. For instance, Chapter II has been used as an introduction to finite lattices, Chapters VI and VII as a course on matroids, and parts of Chapter VII and Chapter VIII as a course on transversal theory and the major existence results. The exercises have been graded. Unmarked exercises can be solved without a great deal of effort; more difficult ones are marked with an asterisk (*). The symbol ~ indicates that the exercise is particularly helpful or interesting, but in no instance is the statement or the solution of an exercise necessary to the development of the subject. The references given at the end are, of course, by no means exhaustive; usually they have been included because they were used in one way or another in the preparation of the text. Books are indicated by an asterisk. The German version of the present book appeared in two volumes-Kombinatorik I. Grundlagen und Ziihltheorie; and II. Matroide und Transversaltheorieas Springer Hochschultexts. Combining these two parts has been a more formidable task than I originally thought. Most of the material has been reorganized, with the major changes appearing in Chapter VIII due to many new results obtained in the last few years. I had the opportunity of working as a research associate at the Department of Statistics of the University of North Carolina in the Combinatorial Year program 1968-1970. It was during this time that I first planned to write this book. Of the many people who have encouraged me since and furthered this work, lowe special thanks to G.-c. Rota, R. C. Bose, and T. A. Dowling for many hours of discussion; to H. Wielandt, H. Salzmann, and R. Baer for their constant support; to R. Weiss, G. Prins, R. H. Schulz, J. Schoene, and W. Mader, who read all or part of the manuscript; and finally to M. Barrett for her impeccable typing. It is my hope that I have been able to record some of the many important changes that combinatorics has undergone in recent years while retaining its origins as an intuitively appealing mathematical pleasure. Berlin September 1979

M. Aigner

Contents

Preliminaries . 1. 2. 3. 4.

Sets .. . Graphs .. . Posets .. . Miscellaneous Notation.

Chapter I. Mappings. . 1. Classes of Mappings 2. Fundamental Orders 3. Permutations 4. Patterns Notes . . . . . Chapter II. Lattices 1. Distributive Lattices. 2. Modular and Semimodular Lattices. 3. Geometric Lattices . . . . . 4. The Fundamental Examples Notes . . . . . . . . . . . . . Chapter III. Counting Functions . 1. The Elementary Counting Coefficients. 2. Recursion and Inversion 3. Binomial Sequences. 4. Order Functions. Notes . . . . . . . . . Chapter IV. Incidence Functions. 1. The Incidence Algebra 2. Mobius Inversion . . . 3. TheMobius Function. 4. Valuations. Notes . . . . . . . . . .

1 2 3 4 5 5

9

20 26

29

30 31 42

52

67 71 73 73 85

99

118 136

137 138

152 166 181

195

viii

Chapter V. Generating Functions. 1. Ordered Structures . . 2. Unordered Structures. 3. G-patterns .. 4. G,H-patterns . . . . . . Notes . . . . . . . . . . Chapter VI. Matroids: Introduction 1. Fundamental Concepts . . 2. Fundamental Examples. . 3. Construction of Matroids. 4. Duality and Connectivity . Notes . . . . . . . . . . . .

Contents

· 196 · 197 · 211

· 220 235 254

255 · 256 270 285 306 321

Chapter VII. Matroids: Further Theory . l. Linear Matroids. . 2. Binary Matroids. . . . 3. Graphic Matroids. . . 4. Transversal Matroids . Notes . . . . . . . . . .

322 322

Chapter VIII. Combinatorial Order Theory . 1. Maximum-Minimum Theorems 2. Transversal Theorems. 3. Sperner Theorems. 4. Ramsey Theorems. Notes, ...

391

391 · 403 · 418

Bibliography . .

· 453

List of Symbols.

· 471

Subject Index. .

· 477

336 351 377

390

.440

· 451

Preliminaries

It seems convenient to list at the outset a few items that will be used throughout the book.

1. Sets We use the symbols N, 7L, 10, IR, and IC for the basic number systems, and set No = {O, 1,2, ... }, Nn = {1, 2, ... , n}; in chapter III the notation n for Nn is also used. ~ij is the Kronecker symbol; idM stands for the identity mapping of a set M onto itself and 2M for the power set of M. The cardinality of a set M is denoted by IMI and we set IMI = 00 whenever M is infinite. For any set M we use the symbol Mk for the cartesian product, Mk = {(aI' ... , ak): ai EM}, and M(k) for the family of k-subsets of M, M(k) = {A !;;; M: IAI = k}. A finite set M with IMI = n is called an n-set. To define a set or a term we use := or :. The following rules are the basic tools for enumeration: (i) Rule of Equality: If Nand R are finite sets and if there exists a bijection between them, then IN I = IR I; (ii) Rule of ~ums: I~{Ai: i E I} i~ a finite family of finite pairwise disjoint sets, then IUiEI Ad - LiEI lAd, (iii) Rule of Products: If {Ai: i E I} is a finite family of finite sets, then for the cartesian product DiEI A;, IDiEI Ad = DiEI lAd· We use the symbols A () B or ();EI Ai to indicate that the sets involved are disjoint. A multiset on S is a set S together with a function r: S --+ No (giving the multiplicity of the elements of S). A convenient notation for a multi set k on S is k = {aka: a E S} with ka := rea), a E S. The usual notions for sets can be carried over to multisets. For instance, if k = {aka: a E S} and I = {d a: a E S} then

k !;;; I: ka

~

la for all

a E S,

k n I: = {amin(ka,la): a E S},

k u 1:= {amax(ka,la): a E S}. Clearly, the family of multisets on a set S forms a lattice under inclusion; furthermore, this lattice is complete.

2

Preliminaries

2. Graphs An undirected graph G(V, E) consists of a non-empty set V, called the vertex-set and a multiset E of unordered pairs {a, b} from V, called the edge-set. A simple graph is a graph that contains no loops {a, a} and no parallel edges {a, b}, {a, b}, i.e., in which E s;;; V(2) is an ordinary set. A directed graph or digraph G(V, E) is a non-empty set V of vertices and a multiset E of ordered pairs (a, b) from V. The elements of E are now called arrows or directed edges. An orientation of an undirected graph G( V, E) is a rule which designates for each edge k = {a, b} a direction (a, b); we then write a = k-, b = e. A graph is finite if both V and E are finite. Except for the definition of a graph itself the terminology follows closely that of Harary [1]. (There, a graph means what we call a simple graph.) The reader is advised to consult chapter 2 in Harary's book for any term not previously defined. We shall, however, redefine most of the notions when they first appear, except for the most basic ones such as connected graph, path, circuit, etc. Whenever we simply use the term "graph" we always mean "undirected graph." Two graphs G(V, E) and G'(V', E') are isomorphic if these exists a bijection ¢: V -+ V' such that {a, b} E E and {¢(a), ¢(b)} E E' appear in E and E' with equal multiplicity. The degree y(v) of a vertex v is the number of edges incident with v where we count loops {v, v} twice. Hence for a finite graph G(V, E) we always have y(v) = 21EI. Two important types of graphs are the complete graphs Kn and the complete bipartite graphs Km,n' Kn is a simple graph with n vertices with any two vertices joined by an edge. K m • n is a simple graph whose vertex-set is the union of two disjoint sets of cardinality m and n respectively, with two vertices being joined if and only if they are in different sets. A bipartite graph is any subgraph of a complete bipartite graph. We shall often denote a bipartite graph by G(Vl U V2 , E) to indicate the defining vertex-sets Vb V2 , where every edge joins a vertex in V1 with a vertex in V2 • The following rule is the single most useful tool in enumeration.

Lvev

(iv) Rule of "counting in two ways": Let G(Vl U V2 , E) be a finite bipartite graph with defining vertex-sets V1 and V2 . Then

L y(v) = L y(v)

veV,

veV2

(= lEI)·

A bipartite graph G(Vl U V2 , E) can also be regarded as a directed graph with all edges directed from Vi to V2 • In other words, bipartite graphs with defining vertex-sets Vi and V2 can be identified with binary relations between V1 and V2 • For this reason, we often use the letter R for the edge-set and in G(V1 U V2 , R) set R(A):= UaeA {y E V 2 : (a, y) E R} for A S;;; Vi' and similarly R(B):= UbeB {XE V1 : (x, b)ER} for B S;;; V2 • For a singleton subset {a}, we simply write R(a). Bipartite graphs have two other important equivalent interpretations. A set system (S, m) is a set S together with a family mof not necessarily distinct subsets of S. Any set system (S, m) gives rise to its incidence graph G(S U m, R) where

3

3. Posets

m:,

(p, A) E R : pEA. Conversely, any bipartite graph G(S u R) yields a set by identifying A E with the set R(A) S;;; S. system (S, A set system (S, can also be described by a 0, l-matrix M = [mij] whose rows and columns are indexed by Sand respectively, with mij = 1 or 0 depending

m:)

m:)

m:

m:

on whether Pi E A j or Pi rt A j. M is called the incidence matrix of (S, m:). Conversely, any 0, l-matrix gives rise to a set system by the reverse procedure.

Example.

2 3 4 5 S

At A2 A3

S = {l, 2, 3,4, 5} m: = {{l,2,4}, {2,5}, {3}, {3,4}}

A4

1 1 0 1 0

0 1 0 0 1

0 0 1 0 0

0 0 1 1 0

m:

Bipartite graph

Set system

0, l-matrix

A graph which has no non-trivial circuits is called aforest. A connected forest is called a tree.

3. Po sets We employ the usual terminology as, for instance, in Birkhoff [1]. If P is a poset then P* denotes the dual poset obtained by inverting the order relation of P. If P contains a unique minimal element, then this element is called the O-element, denoted by 0; similarly, a unique maximal element is called the 1-element, denoted by 1. We say, b covers a or a is covered by b, denoted by a (A)

= (... , Bb , ••• )

The vector (... , Bb'

••• )

. {1 if b E R wIth Bb = 0 if b ¢ R

(A

£;

R).

0

is called the characteristic vector of the subset A.

Clearly, all lattices introduced so far depend up to isomorphism only on the cardinality of R. Hence it makes sense to introduce the symbols Jt(r), 2' and f?4(r). A set R of cardinality r < CI) is called an r-set, and &i(r) the Boolean algebra ofrank r.

Example. Figure 1.4 shows the Boolean algebra 81(4) and its isomorphic lattice where R = {1, 2, 3, 4}.

[~(1)J4,

For brevity of notation, parentheses in the subsets of R were omitted, similarly in the vectors of [~(1)J4. Summarizing our results so far, we see that by comparing images of mappings we obtain the class of products of chains as a first class, fundamental to the whole of combinatorial order theory. In the next chapter we shall describe chain products and their sublattices intrinsically by the distributive property. The various characterizations arising there will then lead to the main questions of combinatorial order theory to be discussed in chapters VI to VIII. B. Refinement

Let us now compare mappings by their kernels. Let f and g be mappings both defined on a set N. Viewing the binary relations ker(f) and ker(g) as subsets of N 2 we may order them by inclusion; thus

f

~

g :~ ker(f)

1234

34

12

o tQ(4)

Figure 1.4

£;

ker(g).

13

2. Fundamental Orders

Again, ~ is reflexive and transitive, but in general not antisymmetric since nothing is said about the images of f and g, respectively. In other words, only the blocks of the kernel partition matter, not the elements onto which the blocks are mapped. Thus we obtain as our second fundamental class the lattices of partitions ofa set N. We shall usually denote partitions by lowercase Greek letters n, (1, p, t, .... By definition n ~ (1 means that any two elements a, b that are in a block of n (or equivalently stand in the equivalence relation (n )-see below) are also in a block of (1. Thus n ~ (1 is equivalent to the statement: every block of n is wholly contained in a block of (1, or, conversely, every block of (1 fully decomposes into blocks of n. For this reason, we say n is a refinement of (1, and call ~ the refinement relation. Other expressions used are n is finer than (1, or (1 is coarser than n. Obviously, the partition consisting of just the single block N is the unique coarsest partition, whereas the finest partition is the one in which all blocks are singletons. Further n - nz , . .. ,nk, 0, 0, ... ) E '!Y can be identified with the number-partitions:

Thus '!Y can be viewed as the lattice of number-partitions ordered by magnitude of parts. The empty partition 0 is the O-element in '!Y and we have nl

+ ... + nk ::; ml + ... + ml k :::;; I and ni :::;; mi (i

= 1, ... , k).

18

I. Mappings

6

11111

5 4

2

n=O

Figure 1.8. The Young lattice

qIj.

Figure 1.8 gives the bottom part of W where for simplicity we use the short notation

n l n2 ... nk for the partition n l + ... + nk. Notice that with every set-partition n ~ AI IA21 ... IAk of an n-set we can associate a number-partition z(n): = n1 + n2 + ... + nk with ni = IAd, i = 1, ... , k. Hence the refinement relation in PJ(n) induces an ordering on the partitions of n:

L ni ::;; L mi:¢> 3n, (J E PJ(n) with n ~ (J and zen) = L n;, z«(J) = L mi· The reader may verify that this in fact gives a po set and that the definition 9f ::;; is equivalent to k

I

i=1

i=1

L ni 1 = k -

1 and the m;'s coincide with the n/s except for one pair

na, nb and one me for which me

=

na

+ nb.

The resulting poset is-called the dominance order 9&(n). The poset 9&(7) is illustrated in Figure 1.9. EXERCISES

I.2

1. Show that every interval of a Boolean algebra &I(S) is again a Boolean algebra. (Hint: For A s; B show [A, B] ~ &I(B - A).)

2. Prove that &I(n) is a complemented lattice for n < 00. That is, for every a E &I(n) there exists a' E &I(n) with a /\ a' = 0 and a v a' = 1. 3. Can ex. 2 be generalized to .A(r), to arbitrary chain products? 4. Show that every interval Em, n]

S;

ff is isomorphic to a finite chain product.

..... 5. Let Land M be lattices. A mapping ¢: L ..... M is called an inf-homomorphism if ¢(x /\ L y) = ¢lex) /\ M¢(Y). A sup-homomorphism is similarly defined.

19

2. Fundamental Orders 7

Sl1

322

1111111 Figure 1.9. The dominance order f7J(7).

Show: (i) Any inf-homomorphism or sup-homomorphism is monotone. (ii) The converse to (i) is false. (iii) For bijections the concepts inf-homomorphism, sup-homomorphism, lattice isomorphism, and order-isomorphism are equivalent. ..... 6. Show that a poset P is a complete lattice if and only if P has a 1-element and inf A exists for all A £ P . ..... 7. A lattice L is called distributive if x A (y v z) = (x A y) v (x A z) for all x, y, z E L. Show that any sublattice of a distributive lattice is distributive; similarly, for the dual lattice and the product of distributive lattices.

8. Let n, a E &'(n) and b(n) and b(a) be the number of blocks of n and a, respectively. Show that b(n A a) + b(n v a) ;:::: b(n) + b(a). ..... 9. Let n, a E &'(n) and G(V, E) be the undirected bipartite graph with the defining vertex-sets corresponding to the blocks Ai of nand Bj of a respectively, and {Ai' Bj } E E if and only if Ai n Bj #- 0. Show: (i) I VI = b(n) + b(a) (ii) lEI = b(n A a) (iii) b(n v a) = # {connected components of G}. (iv) Deduce from (i), (ii), (iii) a new proof of ex. 8.

20

I. Mappings

10. Verify in detail that the intersection of congruence classes is again a congruence class. Describe the supremum of two congruence classes. 11. How many atoms does 2(n, q) have? .s;1(n, q)? 12. A lattice L is called modular if x 1\ (y V z) = (x 1\ y) V z for all x, y, z E L with z ~ x. Prove that %(A) is modular for any group A. -+ 13. Verify that the two definitions of the dominance order are equivalent. -+14. Let n = n 1 n2 ••• nt E rfY. How many elements cover n in rfY; by how many elements is n covered? The same for the dominance order. 15. Let n = n1 n2 ••• nt E rfY. Show that the length of any longest O,n-chain in rfY is Il = 1 ni' What is the corresponding result for the dominance order?

3. Permutations A bijective mapping of a set N onto itself is called a permutation of N. Many problems in counting theory as well as in order theory can be rephrased in terms of a set of permutations acting on the objects in a natural manner. In this section we want to review some basic facts about permutations which will be needed throughout the book. A. Algebraic Properties

The set of permutations of N forms, under composition of mappings, a group called the symmetric group S(N), where the composition g oJ of two permutations g and J is defined by (g J)(a):= g(J(a». The identity in S(N) will be denoted by id. As always we shall mainly be concerned with finite sets N. Since S(N) is determined up to isomorphism by the cardinality of N, we can unambiguously use the shorthand notation Sn. For n < 00, Sn is called the symmetric group ojdegree n and each of its subgroups a permutation group oj degree n; most of the time we choose N = {1, 2, ... , n}. The degree is thus the number of elements being permuted. Fixing {1, ... , n} in the natural order we usually write a permutation 0

as the wordJ(1)J(2)·· ·J(n).

Example. S3 consists of id = 123, 132, 213, 231, 312, and 321, and we have, e.g., = 132 1= 321 = 213.312; thus S3 is not commutative.

312.213

Another common way to representJE Sn is by decomposingJinto cycles

J=

(1,J(l),J2(1), .. .)(i,J(i),J2(i), ...) ....

21

3. Permutations

The order in which the cycles are written down is irrelevant. The length of a cycle is the number of elements contained in it. Cycles of length 1 are called trivial and are often left out in the cycle decomposition off The trivial cycles thus correspond precisely to the fixed points of the permutation. The type off E Sn is the expression

where Mf) is the number of cycles off of length i. It should be remembered that type(f) is just a notational device and not a product. Notice that bl(f) counts the number of fixed points off Clearly n

L Mf) =

n

# {cycles off},

i= 1

L iblf) = n.

i= 1

Example. Let

f=

( 1 2 3 4 5 6 7 8 9)

2 5 6 4 8 7 3 1 9'

Thenf= (1,2,5,8)(3,6,7)(4)(9), and thus type(f) = P3 14l. A permutation f E Sn consisting of one cycle of length k and otherwise only trivial cycles is called k-cyclic. An n-cyclic permutation is called a circular permutation and a 2-cyclic permutation a transposition. Hence the cycle decomposition off is a factorization off into cyclic permutations. A transposition exchanges a pair (i,j) ofletters and leaves the rest unchanged. If N = {1, 2, ... ,n}, then a transposition which exchanges two adjacent elements (i, i + 1) is called a standard transposition. Let G be a permutation group on N. The relation i ~j

:¢>

3g

E

G with j = g(i)

is an equivalence relation on N whose equivalence classes are called G-orbits. Thus, i and j lie in the same orbit if and only if there is a permutation in G which takes i into j. Notice that for a cyclic group G = (f), the G-orbits are precisely the cycles off G is said to be transitive if there is only one G-orbit, namely the whole set N. We denote by O(i) the G-orbit containing i E N and by G(i) the subgroup of G fixing i. G(i) is called the stabilizer of i.

1.6 Lemma. Let G be a permutation group of a finite set N. Then

1000IIG(OI = IGI for all i E N. Proof O(i) consists of all distinct elements g(i), g E G. Since g(i) = h(i) ¢> h -lg(i) = i ¢> h -lg E G(i) ¢> gG(O = hG(i)

22

I. Mappings

we infer that distinct images g(i) give rise to distinct cosets gG(i) and conversely. Hence the cardinalities of these two sets are the same, giving 1O(i) 1= 1G1/1 G(i) I.

D

As a corollary we deduce the following fact, called Burnside's lemma, which will prove important in many counting problems. 1.7 Proposition. Let G be a permutation group on afinite set N. Then

1 k = -IGI

L b (g)

gEG

1

where b 1(g) is the number offixed points of 9 and k is the number of G-orbits. Proof Counting the number of pairs (g, i), 9 E G, i E N with g(i) we obtain

L b (g) = .L IG(i)1 = IGI.L

gEG

1

.EN

.EN

=

i in two ways,

1

10(.)1 = IGlk. D I

Let G and H be permutation groups on sets Nand R, respectively. G and H give rise to another permutation group HG = {h g : 9 E G, hE H} on the set Map(N, R) where

hg(f):= hfg for allfE Map(N, R). We call HG the power group on Map(N, R) induced by G and H. The orbits of HG are thus the equivalence classes under the relation on Map(N, R)

f

~!'

: 3g E G, hE H with!,

= hfg.

We observe that for finite sets N, R the order of HG is IHGI = 1GIIHI (if IR 1> 1) whereas the degree equals 1 Map(N, R)I = IRI'NI.

B. Combinatorial Properties So far we have reviewed some algebraic facts and definitions which will be needed in subsequent chapters. Let us now discuss some combinatorial properties of individual permutations of the set {1, ... , n}. This study of permutations marked the very beginning of systematic research in combinatorial analysis and constitutes, for instance, a major portion of MacMahon's classic treatise [1]. For a detailed account the reader may also consult Foata [3]. We shall concentrate for the rest of this section on permutations f of {1, ... , n} represented as words. An inversion of f = a 1 a2 ... an E Sn is a pair (ai' a) with i aj, and we set

J(f):= {(ai' aj): a;, aj inversion}, i(f):= IJ(f)I.

23

3. Permutations

f is said to be odd or even depending on whether i(f) is an odd or even integer. A descent offis an inversion (a;, ai+ 1) and we denote by DU) the set

of positions where the descents occur. Example. Let f

= 5 6 2 1 7 4 3, then IU)

=

iU)

= 12,

DU)

{(2, 1), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (7, 3), (7, 4)},

= {2, 3, 5, 6}.

Let t = (i, i + 1) be a standard transposition and f = at ... an. Since f· t = at ... ai-tai+ tai ... an we have iU . t) = iU) ± 1 depending on whether i E DU) or not. From this observation we can easily deduce the following result.

nY£)1

1.8 Proposition. Every permutation f E Sn can be written as a product f = tj where the t/s are standard transpositions. Iff = 1 Uj is any other factorization into standard transpositions then k - iU) is an even integer 2::0. In particular, iU) is the minimum number of standard transpositions required to factorize f

n,=

Proof Iff = id then i(f) = 0 and there is nothing to prove. Suppose now f"# id withj E DU), then iU· U,j + 1» = iU) - 1. Hence if we multiply by appropriate standard transpositions tt on the right, the inversion index is reduced to 0, and we obtain f. ti(f)· .. t1 = id. Since t 2 = id for any transposition, it follows that f = tt ... ti(f)· Let f = n~= 1 Uj be any other factorization off into standard transpositions. Then fu k ••• U t = id and iUu k ••• uj ) = iUu k • •• uj +t) ± 1 for all j. Suppose there are exactly p u/s which increase the number of inversions. Since i(id) = 0 this gives i(f)

+p-

(k - p) = 0

or

k - iU) = 2p 2:: O.

0

1.9 Corollary. The standard transpositions generate the whole group Sn.

0

A slight generalization of 1.9 is given in the following statement whose easy proof is left to the reader. 1.10. Let T be a set ofn - 1 transpositions in Sn and let G T be the graph with vertex set {1, ... , n} such that i andj are joined if and only if(i,j) E T. Then T generates Sn if and only if G T is a tree. 0

Another well-known consequence of 1.8 is the following proposition.

24

I. Mappings

1.11 Proposition. For any J, g E Sn

In particular, the set An of even permutationsforms a subgroup ofSn of index 2, called the alternating group An. Proof Letf = n~= 1 til g = n~= 1 Uj be factorizations into standard transpositions. = tl •.• tkul ... U 1 and hence by 1.8

Then fg

i(J) = k - 2p, i(g) = I - 2q i(fg) = k Thus ¢: Sn

--+

+ 1- 2r = i(f) + i(g) + 2(p + q - r).

{1, -1}, given by ¢(!)

= (-1Y(f), is an epimorphism with kernel

0

An.

To conclude, let us derive a characterization of even permutations in terms of their cycle decompositions. Letfbe of type 1n-kk, i.e.,Jis k-cyclic. Applying 1.8 it is readily verified thatfis even if and only if k is odd. In particular, any transposition is odd.

1.12 Proposition. Let f

E

Sn be of type 1b '2 b2 ••• nbn • Then

n

n

-

Lb

i

is even.

i= 1

Proof As f can be written as a product of bi cyclic permutations of length i, i = 1, ... , n, by the remark just made we have n

i(f) ==

L (i -

i= 1

1)b i == b2

+ b4 + . ..

(mod 2)

and n

L (i -

i= 1

EXERCISES

n

l)b i = n -

L bi·

i= 1

0

I.3

1. Show that the order of g E Sn equals the smallest common multiple of the lengths of the cycles of g.

25

3. Permutations

2. Show that a transitive permutation group of degree n > 1 contains an element of order n. -+ 3. Show that the alternating group An is generated by the cyclic permutations (1, 2, i), i = 3, ... , n. 4~ Prove that An contains no non-trivial normal subgroup for n > 4. (Hint: A normal subgroup of An which contains some (1, 2, i) must be An.)

5. Show that in any permutation group G, either all permutations are even or there are equally many even and odd permutations. -+ 6. Let INI = IRI = 3 and G = S3' H = C 3 permutation groups on Nand R respectively. Describe the power group HG. What are the subgroups, cycle-types, etc. ? -+ 7. Prove that every g E Sn, g =1= ld, is the product of disjoint cyclic permutations and that this factorization is unique up to order. -+ 8. Let g E Sn be of type 1n-kk. Show that g is even if and only if k is odd. 9. Prove 1.10. 10~ Let T = {t l ,

... , tn-d be a set of transposItlOns in Sn' Show that g = Oi,:-l ti is a circular permutation if and only if the graph GT in proposition 1.10 is a tree. (Denes)

-+11. Let N = {t, 2, ... , n} and IE Sn be given in word form I = a l a2'" an' We define E(j):= {(a;, a): i < j, ai < aj}. Show thatf ::;:;; g: E(j) ~ E(g) is a partial order on Sn and that this poset is in fact a lattice Per(Sn) on Sn' (Guil baud - Rosenstiehl) -+ 12. Show that in Per(Sn) the following hold: (i) Let IE Per(Sn)' The length of any maximal 0, I-chain is equal to (i) - i(f); the length of any maximal I, 1-chain is equal to i(j). (ii) f : L -+ nt=1 C j induces a partition of P into chains (by running through each coordinate). Hence we have reduced the question as to the minimal dimension of a coding to the following problem in arbitrary finite posets: What is the minimum number d(P) of disjoint chains into which a poset P can be decomposed? d(P) is called the Dilworth number of P. Quite obviously,

d(P)

~

max

IAI,

A antichain

since any two elements of an antichain must appear in different chains. That, in fact, equality holds is one of the fundamental results in all of combinatorics and one of the starting points for the large and still growing field of transversal theory to be discussed in chapter VIII. To return to the example £18(S) notice that since P = S is itself an antichain, we must have d(91(S» ~ IS I = n for any coding; this means that n is in fact the minimal possible dimension.

38

II. Latti a2] contains at most 4 elements. 7': Let L be distributive. (i) Prove that L[al' a2, a3] contains at most 18 elements. (ii) Prove that 3(aI'(3» is generated by 3 elements and has precisely 18 elements. (Hint: The generators are the maximal irreducible elements.) (iii) Let Xl> X2' X3 be the generators of 3{aI'(3» and al> a2, a3 arbitrary elements of some distributive lattice L. Show that the mapping : b 1 b2 ••• bn --. {bl' b2 , ••• , bn } maps the set of injective words in R of length n onto the set of all n-subsets of R. Each n-subset is the image under ¢ of precisely n! injective mappings obtained by permuting the b;'s. The result follows by 3.2. D

75

1. The Elementary Counting Coefficients

The binomial coefficients (~) have no combinatorial meaning for negative val4es of r. But we can formally extend 3.4 to negative arguments by using the definition of lower factorials: (

-r) = [-rJn = (-r)(-r - 1)···(-r - n n n! n! =(-It

The expressions r(r denoted by

+ 1)··· (r + n -

[rJn:= r(r

r(r

+ 1)··· (r + n ,

n.

+ 1) 1)

.

1) are called rising Jactorials oj length n,

+ 1)···(r + n -

1)

with the convention [rJO = 1. Hence we have 3.5.

[-rJn = (-lnrJn [ -rJn = (-lnrJn

(n, rENo) (n, rENo).

This extension of combinatorial identities to all integers and even to real or complex numbers will provide short proofs of many counting formulae. We shall return to this subject in the next section. 3.5, relating the two counting sequences {[rJn: n E No} and {[rJn: n E No}, is the first example of what is called a combinatorial reciprocity law. A study of general reciprocity and inversion theorems will be undertaken in section IY.2. 3.6. The number oj n-multisets in R (i.e., the number oj monotone words oj length n in R) is

IMon(N, R)I

=

err n.

Proof Let N = {I, ... , n} and R = {I, ... , r}, both with the natural order. The mapping cp: b 1 b 2 ••• bn --+ {bl' b 2 + 1, ... , bn + (n - I)} is a bijection between Mon(N, R) and the set of all n-subsets of {I, 2, ... , r + n - I}. 0

Example. Let N = {I < 2 < 3 < 4}, R = {a < b < c}. Then IMon(N, R)I = [3J4/4! = 15, and the words are aaaa aaab aaac

aabb aabc aacc

abbb abbc abcc

accc bbbb bbbc

bbcc bccc cccC.

76

III. Counting Functions

There is also a combinatorial meaning of the rising factorials alone. [r]n counts the number of sortings of n objects into r linearly ordered boxes. (See the exercises) The binomial coefficients are probably the best known of all combinatorial counting numbers. Their applications abound in all branches of mathematics. Let us for the moment just list two basic properties. 3.7 Proposition. Let n EN. The sequence {(k): k = 0, ... , n} of binomial coefficients

satisfies: (i)

(~) = (n ~ k)

(k = 0, ... , n),

(ii)

(~) < G) < ... < (n;2) > (n/2: 1) > ... > (:)

(n even)

(~) < (~) < ... < (n _n 1)/2) = (n +n 1)/2) > ... > (:) (n odd).

0

Property (ii) states that the sequence {(k): k = 0, ... ,n} rises to a maximum (which is possibly attained twice) and falls thereafter. Sequences that rise and fall in this fashion are called unimodal. To be precise: Definition. A sequence Vo, Vl' •.• , Vn of real numbers is called unimodal if there exists an integer M ~ such that

°

We shall verify the unimodality property for other basic sequences in this and the next section. In chapter VIII we shall take the subject up again when we examine the level numbers of finite posets with a rank function.

3.8 Binomial Theorem. Let A be a commutative ring. Then for all a, b E A

Proof. Writing (a + b)n = (a + b)(a + b)··· (a + b) we see that the right-hand side is the sum of all products akbn- k where akbn- k appears as often as we can choose k a's from the n factors, i.e., (k) times. 0

77

1. The Elementary Counting Coefficients

For A

=

t, (i) a = b = 1 and (ii) a =

-1, b = 1 we obtain:

3.9. (i) (n ~ 1).

(ii)

The binomial coefficients admit several useful generalizations. Let us associate with each A ~ N its characteristic function JA: N -. {O, I}, given by

{

JA(a) =

° I

if a E A if a ~ A.

The bijection 4;: 91(N) -. Map(N, {O, I}) with 4;A = JA maps the k-subsets of N precisely onto the mapsJE Map(N, {O, I}) with IJ- 1(1)1 = k. This suggests the following generalization. Consider Map(N, R) and denote by

= {b1> ... ,br }withIJ-l(bj)1 = kj(i = 1, ... ,r).

thenumberofmappingsJ:N -.R The numbers

are called the multinomial coefficients. We note the following generalizations of 3.4, 3.8, and 3.9(i).

3.10 Proposition. We have:

(i)

(

n k1> ... ,kr

n! { ) = k 1 ! k2 !, ... , kr !

°

(iii)

(a 1

+ ... + ar )" =

L

E

L

A. Then

(k, ..... k.)

(k, ..... k.)

1

otherwise.

Let A be a commutative ring, al"'" ar (ii)

j=

(k

(k

n 1>''''

n k

1> ••• ,

r

)a~'a~2 ... t/.•.

k ) =,n. r

78

III. Counting Functions

Proof We prove (i). Suppose n = ki + k2 are mapped onto b i E R can be chosen in

+ ... + kr.

The ki elements which

ways. Assuming that we have picked these elements we can choose the k2 elements which are mapped onto b 2 ERin

ways. Continuing in this way we find that (

n ) = (n) (n - kl) ... (n - ki - ... - kr- I ) kl' ... , kr ki k2 kr n! (n - k l )! kl!(n - k l )! k2!(n - ki - k2)! n! k l !k 2!···kr l"

(n - ki - ... - kr - I )!

kr!(n - ki - ... - kr)!

D

C. The Number of k-subspaces of an n-dimensional Vector Space

Let us turn to vector space lattices !l'(n, q) over finite fields. In 2.64 the number of k-dimensional subspaces of an n-dimensional vector space V over GF(q) was called the Gaussian coefficient (~)q. There are several close analogies between Viewing, for instance, 3.11 the numbers q and the binomial coefficients below as a function of the real variable q, it is easily verified that q tends to as q tends to 1. The same relationship can be observed for most other formulae involving the numbers q •

m

m

m.

m

m

3.11. The number of k-dimensional subspaces of an n-dimensional vector space V(n, q) is (qn _ 1)(qn-1 _ 1) ... (q"-k+ I

-

(qk _ 1)(qk-1 - 1) ... (q - 1)

1)

(k = 0, ... , n).

D

Proof Let us first determine the number Un, k of ordered k-tuples of linearly independent vectors in V(n, q). As the first coordinate of such a k-tuple we can take anyone of the q" - 1 vectors different from O. Any vector v¥:O spans a one-dimensional subspace containing q vectors. Hence there are q" - q vectors linearly independent of v and anyone of them can be taken as the second coordinate. Let w be one of these. The pair {v, w} spans a two-dimensional subspace containing q2 vectors. Hence there are q" - q2 vectors linearly independent of

79

1. The Elementary Counting Coefficients

{v, w} and anyone of them can be taken as the third coordinate. Continuing in this way we conclude that

Each k-tuple of linearly independent vectors spans a k-dimensional subspace and, conversely, any k-dimensional subspace possesses Uk,k ordered bases. Thus we obtain

and the theorem follows by cancelling powers of q. 0

If we write 3.11 in the form 3.12.

D7

and compare this with 3.4 we see that = 1 (qi - 1) is the "q-analogue" of the number n!. Later on we shall translate some other counting coefficients pertaining to sets into their q-analogues pertaining to vector spaces. For the moment, let us note the properties corresponding to 3.7 for the sequence (k)q. 3.13 Proposition. Let n E N. The sequence {(k}q: k = 0, ... , n} of Gaussian coeffi-

cients satisfies:

(k=O, ... ,n),

(i)

(~)q=(n~k)q

(ii)

(~)q < (~)q < ... < (n;2)q> (n/2 n+ l)q > ... > (:)q

(~)q
... ,unontolinearly independent vectors in W. Hence we obtain r" =

=

L (r - l)(r - q) ... (r -

usv

±(k

k=O

n) (r - 1)(r - q) ... (r -

k=O

i (n)k (r if-I) = i (k

q,,-r(U)-I) =

q

n) 9k(r).

k=O

r

1)· .. (r _ q,,-k-l)

q

q

D=o (k)q9k(X) agree in infinitely

As can be any power of q, the polynomials xn and many values of x and hence must be identical. 0

B. Recursions Using our knowledge of polynomial sequences and their connecting coefficients we can now establish some recursion formulae. Our method will always be the same: Any linear functional on R[x] is uniquely determined by the values it assumes on a basis. In particular, for a polynomial sequence {Pn(x)} and a sequence {a,,} of real numbers there is precisely one functional L: R[x] -+ R with LPn(x) = a" for all n. The idea is to translate relations between the polynomials Pn(x) (which are trivial in most instances) into relations between the numbers an' We note the following obvious polynomial recursions.

3.28 Proposition. For all n E 1\1 0 : (i) (ii) (iii) (iv) (v) (vi) (vii)

x n+ 1 = xx" [x]" + 1 = x[x]n - n[x]" [x]" + 1 = x[x - 1]" [x]n+ 1 = x[x]n + n[x]" [x]" + 1 = x[x + 1]" (x - 1)"+ 1 = x(x - I)" - (x - 1)" 9"+ 1(X) = X9n(X) - q"9,,(X). 0

3.29 (Recursion for the Stirling Numbers of the Second Kind). For all n, k (i) So,o = 1, S",o = 0 for n > 0, (ii) S,,+ 1,k = Sn,k-l + kSn,k> (iii) S,,+ 1,k = 1 (j)Sj,k-I'

D=

~

0:

90

III. Counting Functions

Proof. Consider the functionals Lk on

~[x],

k

E

l\Jo, given by

Lk[X]n = bn,k (Kronecker symbol). By 3.24(i) L"xn = L,,(toSn,i[X]i) = itoSn'iL,,[X]i = Sn,k' It follows from 3.28(ii) that

which according to the definition of L" is equivalent to

(+)

(k

~

1).

Since the falling factorials [x]n are a basis of ~[x] we infer by linear extension that (+) holds for all polynomials p(x) E ~[x], i.e., (k ~ 1).

For p(x) = xn this yields

or

To obtain the recursion (iii) we use 3.28(iii). In this case

i.e., (k ~ 1),

and thus for k

~

1 L"_lP(X) = L"xp(x - 1) for all p(x) E

For p(x) = (x

+ It this gives

~[x].

91

2. Recursion and Inversion

From 3.29(i)(ii) we obtain the complete table of the numbers Sn,k' and hence by 3.14 of the numbers ISur(N, R) I as well. 3.29(ii) is an example of a triangular recursion because the three numbers Sn+ I,k' Sn.k-I and Sn,k form a triangle in the Stirling table. 3.29(iii) is a vertical recursion; in this case the second parameter remains fixed. The table below shows the fiTst rows of the table of Stirling numbers of the second kind. The empty cells are to be filled with O's. Stirling numbers of the second kind Sn,k

k=O

n=O 1 2 3 4 5 6 7 8

1 0 0 0 0 0 0 0 0

1

1 1 1 1 1 1 1

1

2

3

4

5

6

7

8

1 3 7 15 31 63 127

1 6 25 90 301 966

1 10 65 350 1701

1 15 140 1050

1 21 266

1 28

1

As a consequence of 3.29 we can say more about the sequence

{Sn,k:

k

= 0, ... , n}.

3.30 Proposition. The sequence {Sn,k: k = 0, , .. , n} is unimodal for all n E No. Let M(n) = max{k: Sn,1 = max}. Then {Sn,k} is of one of the following two types: (i) Sn,O < Sn,l < ... < (ii) Sn,O < Sn,l < ... < Furthermore, M(n)

=

M(n -

Sn,M(n)

>

Sn, M(n) + 1

> .. , >

Sn,n,

= Sn,M(n) > ... > Sn,n' 1) + e(n) with e(n) = 0 or 1.

Sn,M(n)-1

Proof. We use induction on n. For n = 0 or 1 there is nothing to prove. Suppose our proposition holds for i ~ n. Then M(i) ~ MU) for 1 ~ i ~ j ~ n. Let 2 ~ k ~ M(n). Then by 3.29(ii)

and the right hand side is positive by the induction hypothesis, Now suppose M(n) + 2 ~ k ~ n + 1. Then by 3.29(iii)

Here the right hand side is negative because of the induction hypothesis and MU) ~ M(n) for all j ~ n. Hence {Sn,k} is unimodal with M(n + 1) = M(n) or M(n + 1) = M(n) + 1. 0

92

III. Counting Functions

It is not known whether {Sn,k} always has a single maximum for n ~ 3. Some results concerning this problem and the value M(n) have been established (see, e.g., Canfield [1]). Recall the definition of the Bell numbers Bn:

Bn = IY'(n) I =

n

L Sn,k'

k=O

3.31 (Recursion for the Bell Numbers).

Bo = 1 Bn+ 1 = kto

Proof Define the functional L:

~[x] -+ ~

L[X]k

(~)Bk'

by

1 for all kENo.

=

It follows from 3.24(i) that n

Lxn =

L Sn,k = Bn·

k=O

Applying the recursion [X]n+l = x[x - l]n we deduce that

L[x]n+ 1 = L[x]n = Lx[x - l]n, hence

Lp(x) = Lxp(x - 1) for all p(x) E For p(x) = (x

~[x].

+ 1)n this yields

Bell numbers

n=O Bn

1

2

3

4

5

6

7

8

1

2

5

15

52

203

877

4140

The following recursion formulae are derived from 3.28 by using the same arguments as in the proofs of 3.29 and 3.31. Accordingly, except for 3.35 whose proof requires a little more care, we will confine ourselves to listing the results.

93

2. Recursion and Inversion

3.32 (Recursion for the Stirling Numbers of the First Kind). For all n, k (i) so,o = 1, sn,O = 0 for n > 0, (ii) Sn+1,k = Sn k-1 - nSn,b (iii) Sn+ 1,k = Li=o (-l)j[n]jSn_ j,k-l'

~

0:

D

Stirling numbers of the first kind sn,k

k=O

1

2

3

n=O 1 2 3 4 5 6 7 8

1 0 0 0 0 0 0 0 0

-1 2 -6 24 -120 720 -5040

1 -3 11 -50 274 -1764 13068

1 -6 35 -225 1624 -13l32

4

5

6

7

8

-15 175 -1960

1 -21 322

1 -28

1

1

-10 85 -735 6769

It can be shown by an argument similar to that in 3.30, that the sequence

{ Isn, kI: k = 0, . , . , n} is unimodal for every n.

3.33 (Recursion for the Binomial Coefficients). For all n, k

~

0:

(~) = 1, (n + k

1) = ( kn- l)+ (n) k'

D

Binomial coefficients (Pascal's triangle) (;;)

k=O

1

n=O 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8

2

1 3 6

10 15 21 28

3

4

5

6

7

8

1 4 10 20 35 56

1 5 15 35 70

1 6 21 56

1 7 28

1 8

1

94

III. Counting Functions

3.34 (Recursion for the Gaussian Numbers). For all n, k

~

0, q a prime power:

(~t = 1,

(n:

1)q =

(k:

l)q

+ qk(~)q' 0

3.35 (Recursion for the Galois Numbers Gn,q)' For all n

Go,q = 1, Gn + l,q

~

0, q a prime power:

G1,q = 2,

= 2Gn,q + (qn - 1)Gn- 1,q'

Proof Since by definition Gn,q = I!l(n, q)l, Go,q = 1 and G1,q = 2 are clear. Now define the functional L: lR[x] --+ IR by Lgn(x) = 1 for all n, where gn(x) are the Gaussian polynomials. It follows from 3.27(ii) that

Lxn = Gn,q' Hence the asserted recursion is equivalent to

(+)

Lxn+ 1 = 2Lxn + (qn _ 1)Lxn- 1.

Applying 3.28(vii) we have

and thus

(+ +) We introduce an operator Dq on lR[x], setting

D p(x) = p(qx) - p(x) q x Dq is linear and it is immediately verified that

Using Dq we can rewrite ( + +) as

(p(x) E lR[x]).

95

2. Recursion and Inversion

Since {gn(x}} is a basis, this identity holds for all p(x} E Lxp(x}

=

2Lp(x}

~[x],

i.e.,

+ LDqp(x}.

For p(x} = xn we obtain Lxn+ 1

= 2Lxn + LDqxn = 2Lxn + (qn

which is precisely ( + ).

- 1}Lxn-

1

0

For many combinatorial numbers one has to use longer recursions. As an example let us consider the partition numbers Pn,r'

3.36 (Recursion for the Partition Numbers). For all n Pn,l = Pn,n = P n. r

=

Pn-r,l

~

r

~

1:

1,

+

P n - r ,2

+ ... +

P n - r ,,.·

Proof The initial values P n, 1 = P n, n = 1 are clear. Let ex be a partition of n - r into k ~ r parts, say ex: n 1 + ... + nk' We define a new partition ex': (nl + 1) + (n2 + 1) + ... + (nk + I) +) + ... + 1,. ex' is a partition of n into r parts, and

Let {Pn(x}} and {qn(x}} be two polynomial sequences. What we called connecting coefficients can be considered as the elements of the two transformation matrices which transport one basis into the other. The fact that these two matrices are inverses of each other gives rise to inversion formulae for the corresponding connecting coefficients. Let us formulate this as a proposition.

96

III. Counting Functions

3.37 Proposition. Let {Pn(x): n E No} and {qn(x): n E No} be polynomial sequences with connection coefficients an, k> bn,k, i.e., n

qn(x)

=

L an,kPk(X)

k=O n

Pn(x) = If Uo,

UI""

;

Vo,

VI""

L bn,kqk(x)

k=O

are real numbers, then

n

Vn =

L an,kuk (n E No)

k=O

n

L bn,kvk (n E No)·

Un =

¢>

k=O

Proof. The hypothesis implies that the matrices A = [an,k], B = [bn,k] are inverses of each other, where we set an, k = bn , k = 0 for all n < k. But this means V

= Au¢>u = Bv

for all vectors u = (uo, ut> ... ), V = (vo, VI' ... ).

0

Using our knowledge ofthe connecting coefficients between our basic sequences (3.24, 3.25, 3.27) we obtain the following classical inversion formulae. 3.38 Corollary. Let uo , ut> ... ; Vo,

VI' .,.

be real numbers.

(i) Binomial inversion

(ii) Stirling inversion n

vn =

L sn,kuk (n E No)

k=O

n

¢>

un =

¢>

un =

L Sn,kVk (n E No)·

k=O

(iii) Lah inversion n

Vn = (iv) Gauss inversion

L Ln,kuk (n E No)

k=O

n

L Ln,kvk (n E No)·

k=O

97

2. Recursion and Inversion

Proof Let us prove just the binomial inversion formula. From the binomial theorem we have

kt (~)(x

xn =

(x - 1)"

- 1)k,

±(

_1)n-k(n)xk.

=

k=O

k

m

Hence (k), (-1)n- k are the connecting coefficients between the sequences {x n}, {(x - l)n}, and the result follows from 3.37. 0 As an application of the binomial inversion formula we can derive an explicit expression for the Stirling numbers Sn.k' 3.39.

1 ~( - l)k-i(k).n Sn,k = k'.L... . I. I

• 1=0

Proof. By 3.24(i) kn = Jo e)(i!Sn.i) for a fixed n and all kENo· 3.38(i) now yields

k!Sn,k =

±(_I)k-i(~)in.

i=O

EXERCISES

0

I

111.2

1. Show:

(i) L::'=o (k') = (i:! D· (ii) Use (i) and m2 = 2('2) + (1') to prove n

(...) "n III

L...m=O

Lm

m=O

= !n(n + 1)(2n + 1).

m3 -_ ?.

2~ Find a simple expression for

3. Let An = Verify:

2

Lk=O [n]k

3=0

('iD, n < m.

be the number of injective mappings into an n-set.

(i) An = nAn- l + 1, (ii) An = n! Lk=O (ljk!) -+

4. Let In,k be the number of permutations of {I, ... , n} with exactly k inversions (cf. section I.3.B). Prove: (i) I n. o = 1, In. 1 = n - 1, In,k = 0 for k ~ n; (ii) In.(~)-k = In,k; (iii) I n• k = I n,k-l + I n- l ,k for k < n. Derive explicit expressions for In,k' k

=

2,3,4,5.

98

III. Counting Functions

5. Verify the recursion for the Lah numbers:

(i) (ii) (iii) (iv)

L~,o = Lo,o = 1, L~+I,k = L~,k-l + (n

Ln+ l,k = ? Ln,k =

+ k)L~,k'

LJ=o (-1) jsn,jSj,k'

~ 6. A sequence {an} of reals is called log-concave if a; ~ an- l an+1 for all n ~ 2.

Show that {(k)}, {(k)q}, {Sn,k}, {ISn,k I}, and {L~,k} are log-concave for fixed n.

~

7. The Fibonacci numbers Fn are defined by the recursion Fo

= 0,

Fl = 1,

for n

~

2.

Hence: 0,1,1,2,3,5,8,13,21, ... Prove: (i) (ii) (iii) (iv)

Fn- l Fn+1 - F; = ( -1)", Fk IFnk> gcd(Fm' Fn) = Fgcd(m, n), (t(1 + )5»"-2 ~ Fn ~ LPIEY = LP 2EY = L y P 2 =>P I = P 2 • D

=>

LEYP I = LEYP2 => LyP I

3.47 Expansion Theorem. Let P E f/'. Then

where ak = [pXk] x = 0 for all k. Conversely, any expansion Lk~O (aJk!)D k is in f/'. Proof It suffices to consider the sequence {x"}. Write Q = Lk~O (ak/k!)D k with ak defined as in the theorem. Then LPx" = a"

We infer LP = LQ, and thus P = Q by 3.46. D

3.47 suggests the following relationship between f/' and the algebra IJi' of formal power series in one variable endowed with the ordinary product of two series.

107

3. Binomial Sequences

3.48 Proposition. Let !F be the algebra offormal power series over ~ in the variable t. Then

is an algebra isomorphism between !F and f/. It follows, in particular, that any two operators in f/ commute. Proof is clearly injective and also surjective by 3.47. Let f(t) = L~ 0 (aJk !)t", g(t) = L~o (bJk!)t" E!F and set (f) = F, (g) = G. Then (f. g)(t)

= L

(±

,a"bn_" ,)tn

n~O k=O k.(n - k).

= L( n~O

±

1O (aklk!)D k be a Delta operator. We know from 3.5O(iii) that ao = 0, a l ::p O. H-ence by 3.55 there is a unique Delta operator P = P(D) with P(p(D» = p(ft(D» = D.

3.56 Theorem. Let S = s(D), P = p(D) be Delta operators with basis sequences {sn(x): n E No} and {Pn(x): n E No} respectively, and let the operator Z be defined by Z(sn(x»

= Pn(x). Then the following holds:

(i) Z maps any binomial sequence onto a binomial sequence. (ii) !f Z maps rn(x) onto qix) and R = r(D), Q = q(D) are the associated basis operators, then Q = q(D) = r(s(p(D») . .Proof. Z is obviously invertible. Suppose {rix)} is a binomial sequence and that qn(x) = Zrn(x) for all n. Then qo(x) = Zro(x) qn(x)

=

Zrn(x)

= Z1 =

= ZSo(x)

= Po(x) =

ZCt/n,kSk(X»)

1

= kt/n.kPk(X)

and thus n

qlO) = L rn,kPk(O) = 0 k=l

for all n ~ 1.

3. Binomial Sequences

113

Hence {qnCx)} is a normalized sequence. Let Q want to show that Q = ZRZ- 1. We have

=

q(D) be its basis operator. We

(ZR)1 = (QZ)1 and (ZR)rnCx) = Z(nrn-l(x» = nqn-l(X) = Qqn(x) = QZrn(x)

(n ?: 1).

Hence ZR = QZ, i.e., Q = ZRZ- 1 . Applying the same argument to Sand P in place of Rand Q we obtain similarly P = ZSZ-l, and thus p k = ZSkZ-l for all k. Let R = g(S) = Lk2: 1 (ak/k !)Sk be the expansion 3.49 of R in terms of S. (Note that ao = 0, a 1 i= 0.) Then

Q as an expansion of P is therefore shift invariant and hence a Delta operator. Finally, we infer from R = g(S) that r(D) = g(s(D» r(s(D» = g(D), and thus Q = q(D) = g(p(D» = r(s(p(D»).

0

3.57 Corollary. Let P be a Delta operator with basis sequence {Pn(x): n E No}. The basis sequence {Pn(x): n E No} of the operator P is given by

Proof. In 3.56, set S = D, R = P. Then Q = q(D) = P(p(D» = D. This means that the operator Z defined by Z(xn) = Pn(x) satisfies ZPn(x) = xn. Hence Pn(x)

=

Z-lxn

xn = Z-lpn(x)

(n ?: 0)

(n ?: 0).

By 3.42

and the result follows by applying Z- 1 to both sides of this equation. 0 We come to our main theorem.

114

III. Counting Functions

3.58 Theorem (Mullin-Rota). Let {Pn(x): n E No} and {qn(x): n E No} be binomial sequences with connecting coefficients cn,k n

qn(x) = L Cn,kPk(X) k=O and basis operators P = p(D) and Q = qeD) respectively. The polynomials rix) = L~=o Cn,k xk are then the basis sequence corresponding to the Delta operator R = q(fi(D»· Proof Set S = Din 3.56. The operator Z defined by Z(xn) = Pn(x) satisfies Zrn(x) qn(x). Hence Q = qeD) = r(p(D», and thus R = reD) = q(fi(D». D

=

Examples. Let us compute the connecting coefficients Cn,k in xn = Lk=O Cn,k[X]k' The polynomials eix) = L~=o Cn,kxk correspond to the operator Li whence by 3.57 n [Ak n] "L.1 ) en(x = k~O k! X

x=O

x

k

n

=

"

k~OSn'kX

(cf. 3.42).

k

The polynomials en(x) are called the exponential polynomials. Let Z be the operator defined by Z[x]n = xn. Then zxn = en(x) and by the recursion [x]n+ I = x[x - 1]n

Through linear extension we infer ZxE-Ip(x) For p(x)

=

(x

+

=

xZp(x)

(p(x)

E

!R[x]).

1)n this yields a recursion for the exponential polynomials

Since en(1) = Bn (Bell number) we obtain at the point x = 1 the recursion 3.31 for the Bell numbers. Another recursion results from 3.53(i). Since I!l = eD - I we have Li = 10g(I + D), hence Li' = I/(I + D). Applying 3.53(i) we infer en(x) = x(I + D)en-I(x), i.e., (n

Now let us look at [x]n

=

Lk=O Cn,k[X]k'

~

1).

Li = log(I + D), V = I - e- D imply

R = reD) = I - e- log(l+D) = I - (I + D)-I = I - (I - D = D(I - D + D2 + ...) = D(I + D)-I.

+ D2 + ...)

115

3. Binomial Sequences

3.53(ii) yields rn(x) = xU

- k=O L _

n

+ D)nxn-l =

x

±

±

(n)Dkxn-l = (n)[n - l]k xn -k k=O k k=O k

(n) k_ k [n - l]n-k x -

L n

k=O

1)

n! (n k -k' k _ 1 x, .

in agreement with 3.24(ii). Our final theorem gives a compact characterization of binomial sequences, underlining the intimate relationship between binomial sequences and exponential series, a subject which we shall study in depth in chapter V. 3.59 Theorem. A polynomial sequence {pix): n E No} is binomial if and only if

±

n;O>:O

pix) tn n!

=

exg(t)

for some formal power series g(t) = Lk;o>:O gk~ with go = 0, gl -# O. In fact, when the sequence is binomial g(t) = p(t) is the indicator of the operator F. Proof. The identity e(x+y)g(t) = exg(t)eyg(t) implies

and hence, by comparing coefficients, the binomiality of {Pn(x)}. If, on the other hand, {pix)} is binomial we have by 3.57 n

_" Pn (X ) - L, k=O

[p~k

n]

x x= 0 k k' x .

and thus by 3.47

Now apply the isomorphism 3.48.

0

Examples. Let us look once more at the exponential polynomials en(x) = L~=o Sn.k xk . By the theorem just proved

"L" en(x) tn n2:0

n.

=

ex(e -l) . t

116

III. Counting Functions

Comparing coefficients of t n one easily obtains

eix) = e- x

xkkn

L Tr·

k;;,O

•

For x = 1 this yields two formulae for the Bell numbers:

1 kn B =n ek;;,ok!

L-

(Dobinski).

We shall re-derive the first of these identities in chapter V where we interpret the right hand side as the exponential generating function for the Bell numbers. As a final example consider the Laguerre polynomials llx). It is immediate that Lg = Lg for the Laguerre operator Lg. Hence we have

"

In(x) tn = x(t/(t-I)) e . n.

L.,

n;;,O

EXERCISES

III.3

1. Prove 3.40.

2. Verify Abel's identity: For all a E (x ~

~

kt (~)X(X -

+ y)n =

+ ka)"-k.

ka)k-I(y

3. Show that the following operators are shift invariant: (i) B: p(x) ~ J~+ I p(t)dt (Bernoulli operator) (ii) H: p(x) ~ J2TnJ~: e- t2 /2 p(x + t)dt (Hermite operator) (iii) Eu: p(x) ~ t(p(x) + p(x + 1» (Euler operator). 4. Let Br be defined by

f

x+r

Br : p(x) ~ x

In particular, BI

p(t)dt.

= B is the Bernoulli operator. Show:

(i) Br is shift invariant, (ii) Br = (E' - l)EII:!.. Expand Br in terms of I:!. (according to 3.49). ~

5. Verify the first terms in Simpson's formula:

f

X+2

x

(

p(t)dt = 2 I

1:!.2

1:!.4

6

180

+ I:!. + - - -

1:!.6

+-

+ ...

180 -

)

p(x).

117

3. Binomial Sequences

6. Let P be a shift invariant operator and p(k) its k-th derivative. Show:

7. Fill in the details in the proof of 3.54. -+

8. Let {pix)} and {qn(x)} be binomial sequences, and let the operator T be defined by TPn(x) = qn(x). Show that T- 1 exists and that U -+ TUT- 1 is an automorphism of the algebra Y. (Mullin-Rota)

-+

9. Prove 3.59 by expanding Ea in terms of P

=

p(D).

10. Let In(x) = Lk~O In.kxk for all n E I\J where In(x) is the Laguerre polynomial. Formulate the Laguerre inversion formula. 11~

The Bernoulli polynomials bn(x) are defined by

Prove: (i) bn(O) = bn (the Bernoulli number of ex. III.2.13), (ii) bn(x) = Lk (k)bn-kx\ (iii) xn = Lk (k)(n + k - l)-lbk(x). Derive inversion formulae from (ii) and (iii). 12. We consider the analogous situation for vector spaces. Let q be a prime power. The Euler translation tffa is defined by tffap(x) = p(qax ). An operator P is called an Eulerian operator if tffap = q-aptffa for all a E ~ and if px n i= 0 for all n > O. Prove that for an Eulerian operator P deg(Pp(x)) = deg(p(x)) - 1 -+ 13~

(p(x)

E ~[x]).

The sequence {pix)} is called an Eulerian sequence if Po(x) (n

E

=

1 and

I\J).

P is the basis operator corresponding to {Pn(x)} if PPn = (qn - l)Pn-l, and, conversely, Pn(x) is then a basis sequence of P. Prove: (i) if {Pn(x)} is Eulerian, then the basis operator is Eulerian; (ii) if P is an Eulerian operator, then P has a unique Eulerian basis sequence; (iii) the correspondence in (i) and (ii) is a bijection. (Andrews)

118

III. Counting Functions

--+14. Verify that the operator Dq used in the proof of 3.35 is Eulerian. Expand Dq in terms of S. What is the basis sequence?

15. Q is called Euler shift inpariant if QsqG = sqGQ for all a E ~. Prove the statement analogous to 3.49: Let P be Euler shift invariant and Q an Eulerian operator with basis sequence {qn(x)}. Then '\' P = L..

k