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Library or Congrcss Catalogiog-in-Publication Data

Stanley. Richar d P.. [date]

Enumeratlve combinatorics

lncludes lndcxes.

1, Comb~natorial enumeration problems. 1. Title.

QA 164.8.S73 19S6 51 l’.62 86-14

ISElN 0-5~~-0bS~h-5 (y. 1)

- -

It is regrettable that a book, once published and on the way to starting a IXe of

its own, cari no longer bear witness to the painful choices that the author had to

face in the course of bis writing. There are choices that confront the writer of

every book: who is the intended audience? who is to be proved wrong’? who willbe the most likely critic’? Most of us bave indulged in the idle practice of drafting

tables of contents of books we know will never sec the light of day. In some

countries, some such particularly imaginative drafts bave actually been sent to

press (though they may not be included among the author’s list of publications).

In mathcmatics. however, the burden ofchoice faced by the writer is SO heavyas to turn off a11but the most courageous. And of a11 mathematics. combinatorics

is nowadays perhaps the hardest to Write on, despite an eager audience that cuts

across the party lines. Shall an isolated special result be granted a section of its

own’? Shall a fledgling new theory with as yet sparse applications be gingerly

thrust in the middle of a chapter? Shall the author yield to one of the contrary

temptations of recreational math at one end, and categorical rigor at the other?

or to the highly rewarding lure of the algorithm?Richard Stanley has corne through these hurdles with flying colors. It has

been said that combinatorics has too many theorems, matched with very few

theories; Stanley’s book belies this assertion. Together with a sage choice of the

most attractive theories on today’s stage, he blends a variety of examples demo-

cratically chosen from topology to computer science, from algebra to complex

variables. The reader will never be at a loss for an illustrative example, or for a

proof that fails to meet G. H. Hardy3 criterion of pleasant surprise.

His choice of exercises will at last enable us to give a satisfying reference to

the colleague who knocks at our door with his combinatorial problem. %ut best

ofall, Stanley has succeeded in dramatizing the subject, in a book that will engage

from start to finish the attention of any mathematician who will open it at page

one.

Gian-Carlo Rota

V



areas, this situation arises quite frequently. As a result, 1 have made a special

effort in this book to include coverage of topics from enumerative combinatorics

that arise in other branches of mathematics.

The exercises found at the end of each chapter play a vital role in achievingThe#% = exere&e s ( %+y;with dif#%X&~ r&ings

of 1 to 3 -) may be attempted by students using this book as a text; the more

difficult exercises are not really meant to be solved (though some readers will

undoubtedly be unable to resist a real challenge), but rather serve as an entry

into areas that are not directly covered by the text. 1 hope that these more difficult

exercises will convince the reader of the depth and the wide applicability of

enumerative combinatorics, especially in Chapter 3, where it is by no means L

priori evident that partially ordered sets are more than a convenient bookkeepingdevice. Solutions or references to solutions are provided for almost a11 of the

exercises.

The method of citation and referencing is, 1 hope, largely self-explanatory.

Al1 citations to references in another chapter are preceded by the relevant chapter

number. For instance, [3.16] refers to reference 16 in Chapter 3. 1 have included

no references to outside literature within the text itself; a11such references appear

in the Notes at the end of each chapter. Each chapter bas its own list of references.

while the references relevant to an exercise are given separately in the solution

to that exercise.

Many people have contributed in many ways to the writing of this book.

Special mention must go to G.-C. Rota for introducing me to the pleasures of

enumerative combinatorics and for his constant encouragement and stimulation.

1 must also mention Donald Knuth, whose superb texts on computer scienceinspired me to include a wide range of solved exercises with a difficulty level

prescribed in advance. The following people have contributed valuable sugges-

tions and encouragement, and 1 thank them: Ed Bender, Lou Bil]era, Anders

Bjorner, Thomas Brylawski, Persi Diaconis, Dominique Foata, Adriano Garsia,

Ira Gesse], Jay Goldman, Curtis Greene, Victor Klee, Pierre Leroux, and Ronald

C. Mullin. In addition, the names of many whose ideas 1 have borrowed are

mentioned in the Notes and Exercises. 1 am grateful to a number of typists for

their lïne preparation of the manuscript, including Ruby Aguirre, Louise Bal-

zarini, Margaret Beucler, Benito Rakower, and Phyllis Ruby. Finally, thanks to

John Kimmel of Wadsworth & Brooks/Cole Advanced Books & Software for

his support a nd encouragement throughout the preparation of this book, and to

Phyllis Larimore for her careful editing.

For lïnancial support during the writing of this book 1 wish to thank the

Massachusetts Institute of Technology, the National Science Foundation, and

the Guggenheim Foundation.

Richard Stanley

Contents

‘Votatiou

Chupter What 1s Enumerative Combinatorics?

1.1 How to Count

1.2 Sets and Multisets

1.3 Permutation Statistics

1.4 The Twelvefold Way

Notes

References

A Note a bout the Exercises

Exercises

Solutions to Exercises

Chapter 2

2.1

2.2

2.3

2.4

2.52.6

2.1

Sieve Methods

Inclusion-Exclusion

Examples and Special Cases

Permutations with Restricted Positions

Ferrers Boards

F-partitions and Unimodal SequencesInvolutions

Determinants

Notes

References

Exercises

Solutions to Exercises



C/zq~ 3 Partially Ordered Sets

3.1 Basic Concepts

3.2 New Posets from Old

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

Distributive Lattices

Chains in Distributive Lattices

The Incidence Algebra of a Locally Finite Poset

The Mobius Inversion Formula

Techniques for Computing Mobius Functions

Lattices and Their Mobius Algebras

The Mobius Function of a Semimodular Lattice

Zeta Polynomials

Rank-selection

R-labelings

Eulerian Posets

Binomial Posets and Generating Functions

An Application to Permutation Enumeration

Notes

References

ExercisesSolutions to Exercises

96

96

100

v

105

110

113

116

117

124

126

129

131

133

135

140

147

149

152

153174

Chapter 4

4.1

4.2

4.3

4.4

4.5

4.6

4.1

Rational Generating Functions 202

Rational Power Series in One Variable 202

Further Ramilïcations 204

Polynomials 208

Quasi-polynomials 210

P-partitions 211

Linear Homogeneous Diophantine Equations 221

The Transfer-matrix Method 241

Notes 260References 263

Exercises 264

Solutions to Exercises 275

Appendix Graph Theory Terminology 293

Index 296

Notation

complex numbers

nonnegative integers

positive integers

rational numbers

real numbers

integers

the set { 1,2,. . . ,rz , for 11 N (SO 10] = @)

for integers i 5 j the set {i. j + 1.. . .j:

greatest integer 5 Y

least integer >xa11 used for the number of elements of the lïnite set X

theset {ui,..., Ut} c R, where a1 < ... < Ut

the Kronecker delta, equal to 1 if i = j and 0 otherwise

equals by delïnition

image of the function A

kernel of the homomorphism or linear transformation A

trace of the linear transformation A

the lïnite lïeld (unique up to isomorphism) with q elements

direct sum of the vector spaces (or modules, rings, etc.) F

ring of polynomials in the indeterminate x with coeflïcients

in the integral domain R

ring of rational functions in x with coefficients in R (R(x) isthe quotient lïeld of R[x] when R is a field)

ring of forma1 power series In>0 U~X” n x with coefficientsu,, in R

ring of forma1 Laurent series 1” k q~ u”Y, for some ne E Z, inx with coefficients un in R (R((x)) is the quotient lïeld ofR [ [x]] when R is a field)

x



CHAPTER 1

_tie

Combinatorics?

1 1 How to Count

The basic problem of enumerative combinatorics is that of counting the number

of elements of a finite set. Usually we are given an infinite class of IÏnite sets Si

where i ranges over some index set 1 (such as the nonnegative integers N). and

we wish to Count the number j(i) of elements of each Si “simultaneously.*’

Immediate philosophical diflïculties arise. What does it mean to “Count” the

number of elements of Si’? There is no delïnitive answer to this question. Only

through experience does one develop an idea of what is meant by a “deter-

mination” of a counting function j-(i). The counting function j(i) cari be given

in several standard ways:

1 The most satisfactory form of j(i) is a completely explicit closed formula

involving only well-known functions, and free from summation symbols. Only

in rare cases will such a formula exist. As formulas for j-(i) become more

complicated, our willingness to accept them as “determinations” of j(i) decreases.

Consider the following examples.

1 1 1 Example For each n E N, let j(n) be the number of subsets of the set

[n-j = {1,2,..., n}. Then j(n) = 2”, and no one will quarrel about this being a

satisfactory formula for j(n).

1.1.2 Example. Suppose n men give their n hats to a hat-check person. Let j”(n)be the number of ways that the bats cari be given back to the men, each man

receiving one bat, SO that no man receives his own hat. For instance, j(l) = 0,

f(2) = 1, j(3) = 2. We will see in Chapter 2 that

j-(n) = n i~f(-l)i/i .

This formula for j(n) is not as elegant as the formula in Example 1.1.1, but for

lack of a simpler answer we are willing to accept (1) as a satisfactory formula. In

1



L CtYJpter What s Enumerati\e Combinatorics?

fact. once the derivation of (1) is understood (using the Principle of Inclusion

Exclusion), every term of (1) bas an easily understood combinatorial meaning.This enables us to “understand” 1) intuitively, SO our willingness to accept it is

enhanced. We also remark~tmat~ it follows easily from (Il that J’(H) is_ hee

intcger to n /e. ThÇs ïs certaïmy a simple explicit formula. but it bas the dis-

advantage of being *‘non-combinatorial”; that is, dividing by e and rounding off

to the nearest integer bas no direct combinatorial significance.

1.1.3 Example. Let ,Jrr) be the number of 11 x rz matrices M of zeros and ones

such that every row and column of M bas three ones. For example, J’(O) = _Q1) =

j’(2) = 0, ,f(3) = 1. The most explicit formula known at present for f(n) is

where the sum is over a11 11 + 2)(/1 + l),? solutions to 2 + /1 + ;J = jr n non-

negative intcgers. This formula givcs very little insight into the behavior of J(ti).

but it does allow one to compute ,f(t~) much fastcr than if only the combinatorial

delïnition of j’(tr) were used. Hence with some reluctance we accept (2) as a

“determination” of f(fr). Of course if someone were later to prove ~(II) =

~I(N I)()l - 2)/6 (rather unlikely), then our enthusiasm for (2) would be con-siderably diminished.

1.1.4 Exampie. There are actually formulas in the literature (“nameless here

for evermore”) for certain counting functions ~(II) whose evaluation rcquireç

listing a11 (or aimost ah) of thc f(tz) abjects being counted Such a “formula.’ iscompleteiy worthless.

2. A rccurrence for j’(i) may be given in terms of previously calculated.f( j)+s,

thereby giving a simple procedure for calculating f(i) for any desired i E . For

instance, let ,f(n) be the number of subsets of [n] that do not contain two

consecutive integers. For example, for n = 4 we have the subsets @, {1 , {2), {3;.

[4;., $31, {1,4J, {2,41, SO f(4)= 8. It is easily seen that J”(a) = f(n - 1) +,f(n - 2) for n 2 2. This makes it trivial, for example, to compute f(20). On the

other hand, it cari be shown that

where T = * l + fi), T = i(l - 3). This is an explicit answer, but because it

involves irrational numbers it is a matter of opinion whether it is a better answerthan the recurrence f(n) = _/(a - 1) + _/(n - 2).

3. An estimate may be given for f(i). If 1 = IV, his estimate frequently takes

the form of an us~~m~r&cfirrrz~/u~(~) - g(n), where g(n) is a “familiar function.”

The notation f(n) - g(n) means that lim”_ f(n)/g(n) = 1. For instance, let f(n)be the function of Example 1.1.3. It cari be shown that

Rr) - e -z36-n(3n) .

For many purposes this estimate is superior to the “explicit” formula (2).

1.1 How to Count 3

4. Thc most useful but most diflïcult to understand method for cvaluating

J(i) is to give its ~~~~~~~~~ri~~~~,~~~~~~~j~~.e will not dcvelop in this chapter a rigorous

abstract theory of generating functions, but will instead content ourselves with

an “abject” that represents a counting function j’(i). Usually this abject is a $NY~~LZ/

puver seritx The two most common types of generating functions are orditwyy~

generatit~~g /imrious and uponentiui genetxting fiuwtions. If 1 = ,Y . hen the

ordinary gencrating function of J()r) is the forma1 power series

while the exponential generating function of J(ri) is the formai powcr stries

(If 1 = p. thc positive integers, thcn these sums begin at 11= 1.) These power

series are callcd “formai*’ because we are not concerned with lctting Y takc

on particular valtics. and we ignore questions of convergence and divergence.

The term _Y”or .Y” i merely marks the place where ~(II) is written. If F(_~I =

x,,>,, o,,.Y”. we cal1 u,, the cwjficimt of _Y” n F(.Y) and Write

Similarly ke cari deal Lvith gcnerating functions of scveral variables. such as

(which may be considered as “ordinary” in the indices 1. rr and “exponential” in

n), or even of infinitely many variables. In this latter case every term should

involve only lïnitely many of the variables.

Why bother with generating functions if they are merely another way of

writing a counting function? The answer is that we cari perform various natural

operations on generating functions that have a combinatorial signifïcance. Forinstance, we cari add two generating functions (in one variable) by the rule

Similarly, we cari multiply generating functions according to the rule

where Ci = xy=,, uibn_i r

1.1 O~ to ount 3



where & = L;E0 (y)~$,,-~, with (y) = ~ /tt (tz - i) . Note that these operations arejust what we would obtain by treating generating functions as if they obeyed the

ordinary laws of algebra, such as _xixj = xi+j. T’hese operations coincide with theaddition and multiplication of functions when the power series converge for

?Il@roll%iaZ~vaIues of~~~~anu ney ooey such famîlia~rlaws of algëbra as associa-

tivity and commutativity of addition and multiplication, distributivity of multi-

plication over addition, and cancellation of multiplication (i.e., if F(~)G(X) =

F(x)H(x) and F(x) # 0, then G(x) = H(x)). In fact, the set of a11 forma1 powerseries 1” z 0 u”x” with complex c oefficients un forms a (commutative) integral

domain under the operations just delïned. This integral domain is denoted by

C [ [x]]. (Actually, C [[.Y]] is a very special type of integral domain. For readers

with some familiarity with algebra, we remark that C[[x]] îs a principal idealdomain and therefore a unique factorization domain. In fact, every ideal of

C[[x]] has the form (.Y”) for some n 2 0. From the viewpoint of commutative

algebra, C[[x]] is a one-dimensional complete regular local ring. These general

algebraic considerations will not concern us here; rather we will discuss from

an elementary viewpoint the properties of C[[xJ] that will be useful to us.)

Similarly, the set of forma1 power series in the rn variables xi, . . xm (where m

may be infinite) is denoted C[ [x l . . . .Y ]] and forms a unique factorizationdomain (though not a principal ideal domain for m 2 2).

It is primarily through experience that the combinatorial signitïcance of thealgebraic operations of C [ [.Y]] or C [ [xi,. . . x ]] is understood, as well as the

problem of whether to use ordinary or exponential generating functions (or

various other kinds discussed in later chapters). In Section 3.15, we will explain

to some extent the combinatorial signifïcance of these operations, but even thenexperience is indispensable.

J

If F(x) and G(x) are elements of C[[?C]] satisfying F(x)G(x) = 1, then we

(naturally) Write G(x) = F(x)-‘. (Here 1 is short for 1 + Ox + Ox2 + ....) It is

easy to see that F(x)-’ exists (in which case it is unique) if and only if u0 # 0,

where F(x) = 1 “z ,, u”x”. One commonly writes “symbolically” u0 = F(O), even

though F(x) is not considered to be a function of x. If F(O) # 0 and F(~)G(X) =

H(x), then G(x) = F(x))lH(x). More generally, the operation -i satislïes a11 he

familiar laws of algebra, provided it is only applied to power series F(x) satisfying

F(O) # 0. For instance, (F(~)G(X))-’ = F(x)-‘G(x)-‘, (F(x)-r)-t = F(x), and SO

on. Similar results hold for C[[xt . . . xn J].

1 1 5 Example Let (~~2,,~“x”)(l - CYX) ~~2,,c#, where a is a non-zero

complex number. Then by the detïnition of power series multiplication,

1

1, rr=ocn =

a” - a(a”-l) = 0, n 2 1.

Hence En>0 a”x” = (1 - ax)-‘, which cari also be written

X1

anxn c ~PI2 1 -ax’

This formula cornes as no surprise; it is simply the formula (in a forma1 setting)

for summing a geometric series.

Example 1.1.5 provides a simple illustration of a general principle that,

informally speakinz. states that if we have an identity involving power series

that is valid when the power series are regarded as functions (SO that the

variables are suffïciently small complex numbers), then this identity continues to

remain valid when regarded as an identity among forma1 power series, prokfed

the operations involved in the formulas are well-dehned for forma1 power series.

It would be unnecessarily pedantic for us to state a precise form of this principle

here, since the reader should have little trouble justifying in any particular casethe forma1 validity of our manipulations with power series. We will give several

examples throughout this section to illustrate this contention.

1 1 6JXxample The identity

(3)

is valid at the function-theorctic level (it states that eXeeX = 1) and is well-defined

as a statement involving forma1 power series. Hence (3) is a valid forma1 power

series identity. In other words (equating coeflïcients of x”#‘n on both sides of (3))

we have

To justify this identity directly from (3), we may reason as follows. Both sides of

(3) converge for a11 K6 :C, SO we have

But if two power series in x represent the same function J(x) in a neighborhood

of 0, then these two power series must agree term-by-term, by a standard

elementary result concerning power series. Hence (4) follows.

1 1 7 Example The identity

“;. (x + lY/rl = e”Fo x”/rI

is valid at the ,function-theoretic level (it states that eX+’ = e.eX), but does notmake sense as a statement involving forma1 power series. There is no formuf

procedure for writing En>0 (x + ly/rr as a member of C[[x]].Although the expression xnzo (x + l)“/n does not make sense fir&ly,

there are nevertheless certain infinite processes that cari be carried out formally

in C [ [xl]. (These concepts extend straightforwardly to C [ [xr . . x J] but

for simplicity we consider only C[[x]].) TO deline these processes, we need to

put some additional structure on C[[x]]-namely, the notion of conuergence.



From an algebraic standpoint, the delïnition of convergence is inherent in the

statement that C[[.Y]] is complete in a certain standard topology that cari be

put on C[[X]]. However. we will assume no knowledge of topology on the part

of the reader and will instead give a self-contained, elementary trgstment ofcon vcïgeEeC

If-FI(x), F2(.y), . is a sequence of forma1 power series, and if F(X) =

~“20a,$’ is another forma1 power series, we say by delïnition that Fi(x) con-

uerges to F(X) as i -t x. wfitten Fi(~) -+ F(X), provided that for a11 n 2 0 there isa number (j(n) such that the coeflïcient of _Y” n Fi(x) is an whenever i 2 J(tz). In

other words, for every n the sequence

[ F1 (.y), [ F* (.Y), .n n

of complex numbers eventually becomesdefinition of convergence is the following,

constant with value a”. An equivalent

Deline the @ree of a non-zero forma1power series F(X) = 1 n2,, a,,.~“, denoted degF(x), to be the least integer n suchthat un # 0. Note that deg F(~)G(X) = deg F(x) + deg G(X). Then Fi(x) convergesif and only if limi_E deg(F,+,(x) - Fi(_~)) = co.

We now say that an infinite sum 1. ,20f+) has the value F(X) providedthat x>+, Fi(x) -+ F(+x). A similar definition is made for the infinite product

nj> l Fj(.yl. TO avoid unimportant technicalities we assume that in any infinite

product ~j~ 1 4(x), each factor q(x) satislïes c(O) = 1. For instance, let e(x) =

aj-x’. Then for I 2 u, the coeflïcient of X” in E;=u Fj(x) is a,,. Hence xjZO Fj(.y) is

just the power series 1 ,,20 a&‘. Thus we cari think of the forma1 power series

xnZ0 a,,.~” as actually being the “sum” of its individual terms. The proofs of the

following two elementary results are left to the reader.

1.1.8 Proposition. The inlïnite series 1. ,ZQ &(x) converges if and only iflimj_ deg e(x) = CC. 0

1.1.9 Proposition. The infïnite product nj2 1 1 + e(x)), where q(O) = 0, con-

verges if and only if limj_m degq(x) = a. 0It is essential to realize that in evaluating a convergent series xj20 4(x) (or

similarly a product fl. ,> l q(x)), the coefficient of x” for any given n cari be

computed using only jhite processes. For if j is suflïcientiy large, say j > d(n),then deg4(x) > n, SO hat

The latter expression involves only a finite sum.

The most important combinatorial application of the notion ofconvergence

is to the idea ofpower series compositon. If F(x) = JTnzO a,,~”and G(x) are forma1

power series with G(O) = 0, deline the composition F(G(x)) to be the infinite sum

xn20 ~“C(X)“.Since deg G(x)” = n.deg G(x) 2 n, we see by Proposition 1.1.8 that

F(G(x)) is well-defïned as a forma1 power series. We also see why an expres-

sion such as el’X does not make sense forrnally; namely, the inlïnite series

~nx,U + $‘/n does not converge in accordance with the above delïnition. On

the other hand, an expression like eeXmlmakes good sense fotmally, since it has

the form F(G(x)) where F(x) = xnkOx”;n and G(x) = x,,> l x”/n .

where (i) = 2(2. - 1)...(2 - n + l)/f~ . In fact, we may regard 2 as an indeter-

minate and take (5) as the definition of (1 + F(.K))’ as an element of C[[.Y,~_]]

(or of C[;_l [[.Y]]; that is, the coefficient of .Y in (1 + F(.Y))’ is a po/JXomia/in 2). Al1 the expected properties of exponentiation are indeed valid. such

as (1 + Fe))i+fl = (1 + F(.Y))‘(~ + F(.Y))~ (regarded as an identity in the ring

C[[X,~,A~]], or in the ring C[[X]] where one takes j.. ~CC\.

If F(x) = x,,kO a,,.?, deline the jtirmal deriuatire F’(.Y) (also denotcd ‘L or

DF(x)) to be the forma1 power series xn2,, MZ”.Y”-’ = x,,ZO(~j + ~)u,,+~.Y”. t is

easy to check that a11 he familiar laws of differentiation that are well-defined

formally continue to be valid for forma1 power series. In particular

(F + G)’ = F’ + G’

(FG)’ = F G + FG’

F(G(x))’ = G’(x)F’(G(x)).

We thus have a theory of fortna/ calchs for forma1 power series. The usefulnessof this theory will become apparent in subsequent examples. We first give an

example of the use of the forma1 calculus that should shed some additional light

on the validity of manipulating forma1 power series as if they were actual

functions of x.

1.1.11 Example. Suppose F(O) = 1, and let G(x) be the unique power series

satisfying

G’(X) = F’(x)/F(x) , G(O) = 0. (6)

From the function-theoretic viewpoint we cari “salve” (6) to obtain F(x) =

exp G(x), where by definition exp G(x) = En>,, G(x)“/~ . Since G(O) = 0 every-

thing is well-defïned formally, SO 6) should remain equivalent to F(x) = exp G(X)

even if the power series for F(x) converges only at x = 0. How cari this assertion

be justified without actually proving a combinatorial identity? Let F(x) = 1 +

xn2 l u”x”. From (6) we cari compute explicitly G(x) = xRZ l &x”, and it is

quickly seen that each b” s a polynomial in lïnitely many of the ai’s. It then follows

that if exp G(x) = 1 + En2 l cnx”, then each c” will also be a polynomial in finitely

many of the ai’s, say c” = P~(U~, *, . . . , a,,,),where m depends on n. Now we know

that F(x) = expG(x) provided 1 + x”21 U”X’ converges. If-two Taylor series

convergent in some neighborhood of the origin represent the same function, then

their coeflïcients coincide. Hence a” = ~,,(a~,u2,. . . , a,,,) provided 1 + En2 I U~X”

&.* . . . . . . ” _ ..



converges. Thus the two polynomials un and pJui,+,. . . ,u,,,) agree in some

neighborhood of the origin of C”, SO they must be equal. (It is well-knownthat if two complex polynomials in m variables agree in some open set of Cm,

then they are identical.) Since un = p,,(~i, u2,. . . , u,,,)as polynomials, the identity#%KG, ,%Z,..,a.. .%...--J”, ,,cC+*Vw”b* C?C,*V>.

There is an alternative method for justifying the forma1 solution F(X) =

exp G(x) to (6), which may appeal to topologically inclined readers. Given G(X)

with G(O) = 0, deline F(x) = expG(x) and consider a map 4 : C[[x]] -+ C[[_Y]]

delïned by #(G(x)) = G’(x) - $$. One easily vernies the following: (a) if G

converges in some neighborhood of 0 then &G(x)) = 0; (b) the set 9 of a11 power

series G(.x) E C[[x]] that converge in some neighborhood of0 is dense in C[[S]],

in the topology delïned above (in fact. the set C[x] of polynomials is dense); and(c) the function 4 is continuous in the topology delïned above. From this it

follows that C$(G(X))= 0 for a11 G(.x)EC[[.X]] with G(O) = 0.

We now present various illustrations in the manipulation of generating

functions. Throughout we will be making heavy use of the principle that forma1

power series cari be treated as if they were functions.

1.1.12 Exampk. Find a simple expression for the generating function F(x) =

xn,0 u,$, where u,, = ~1, = 1, un = un-, + u,,_* if n 2 2. We have

= 1 + .K + _K(F(.K) 1) + x2F(x).

Solving for F(x) yields F(x) = l/( 1 - .Y x’).

1.1.13 Exampie. Find a simple expression for the generating function F(X) =

~,,~ou&‘/~ , where u. = ai = 1, un = a”_i + (n - l)~“_~ if n 2 2. We have

F(x) = 1 U”X”/?ZIl2

= 1 + x + 1 U”X”/r7“22

= 1 + x + 1 (un-1 + (n - l)un_&c”/n .“22

(7)

Let G(x) = 1 ,,2Z un_i.?/~ and H(x) = znkZ (n - l)~~-~x”/n . Then G’(x) =xnkZ ~~_ix”-‘/(n - l) = F(x) - 1, and H’(x) = xn21 u,,-~x~-‘/(~ - 2) =

xF(x). Hence if we differentiate (7) we obtain

F(x) = 1 + (F(x) - 1) + xF(x) = (1 + x)F(x).

The unique solution to this differential equation satisfying F(O) = 1 is F(x) =

exp(x + 4~~). (As shown in Example 1.1.11, solving this differential equation is

a purely forma1 procedure.)

1.1.14 Example. Let P(~I) be the Mobius function of number theory; that is,

~(1) = 1, p(n) = 0 if n is divisible by the square of an integer greater than one,

and p(n) = (- ly if n is the product of r distinct primes. Find a simple expression

for the power series

F(x) = n (1 - xn)-p(n”fl. (8)II>I

First let us make sure that F’(x) is well-detïned as a forma1 power series. We have

by Example 1.1.10 that

Note that (1 - .Y”)-p’n~‘n 1 + H(x), where deg H(x) = n. Hence by Proposition

1.1.9 the inlïnite product (8) converges, SO F(x) is well-defined. Now apply log to

(8). In other words, form IB$ F(x), where

log(1 + x) = t (- lY-‘x”/n,“2

the power series expansion for the natural logarithm. We obtain

log F(x) = log n (1 - x”)-fl(““”“21

= & log( 1 - x”)-PC”“”

= -“zi $)log(l - _Xfl)

= -“;lpJq .The coeflïcient of xm in the above power series is

;F /44,In

where the sum is over a11 positive integers dividing m. It is well-known that

&&l) c ” mIn 0, otherwise.

Hence logF(x) = x, SO F(x) = ex. Note that the derivation of this miraculous

formula involved only for~~u/ manipulations.

1.1.15 Exampie. Find the unique sequence u. = 1, ai, uZ, . . . of real numbers

satisfying

i U&z”_~= 1k=O

(9)

for a11 n E lV. The trick is to recognize the left-hand side of (9) as the coeflïcient



1 Chapter 1 What 1s Enumerative Comhinatorics?

U,, = ( - 1y(_+) = (_ ,Y( -;)( -;)(-;j...( -y)

Il n

l 3 .5 ‘. (2t1 - 1)

2%

NO~ that we have discussed thc manipulation of forma1 power series. thequestion arises as to the advantages of using generating functions to represent a

counting function j(tt). Why, for instance. should a formula such as

.

be regarded as a “determination” of ,f(/t)? Basicaily, the answer is that there are

many standard, routine techniques for extracting information from generating

functions. Generating functions are frequently the most concise and eftïcient way

of prescnting information about their coefficients. For instance, from (10) an

experienced enumerativc combinatorialist cari tel1 at a glancc the fohowing:

1. A simple recurrence for ,f(tz) cari be found by differentiation. Namely. we

obtain

n;. /‘(tz).K”-‘/yn - l) = t1 + x)e-r+‘x--, = ( 1 + .y) & j(tl).Y”.‘tl .

Equating coefficients of Y’n yields

.fCtl+ 1) = _f(tz) + nf(n - l), n 2 1.

2. An explicit formula for f(n) cari be obtained from J?+(~“~) = c’&~.

Namely,

SO hat

3. Regarded as a function of a complex variable, exp(.x + g) is a nicely-

behaved entire function, SO hat standard techniques from the theory of asymp-

totic estimates cari be used to estimate f(n). As a first approximation, it is routine

1.1 HO~ to Count 11

(for someone suflïciently versed in complex variable theory) to obtain the asymp-

totic formula

No other method of describing ~(II) makes it SO easy to determine these

fundamental properties. Many other properties ofj’(n) cari also be easily obtained

from the generating function: for instance, we leave to the reader the problem of

evaluating, essentially by inspection of (lO), the sum

Thercfore vve are ready to accept dhe generating function exp(.y + q) as a satis-

factory determination of f(n).

This completes our discussion of generating functions and more generally

the problem of giving a satisfactory description of a counting function ,/(tr). We

now turn to the question of what is the best way to pr~~~c hat a counting function

bas some given description. In accordance with the Princip]e from other branches

of mathematics that it is bctter to exhibit an explicit isomorphism between two

abjects than merely to prove that they are isomorphic. we adopt the general

principle that it is better to exhibit an cxplicit onc-to-one correspondence (bijec-

tion) between two Imite sets than merely to prove that they bave the same number

of elements. A proof that shows that a certain set S bas a certain number ttt of

elements by constructing an explicit bijection between S and some other set thatis known to have ttl elements is called a cotnbinatorial prmf or bijectioe proof.

The precise border between combinatorial and non-combinatorial proofs is

rather hazy. and certain arguments that to an inexperienced enumerator will

appear non-combinatorial will be recognized by a more experienced counter as

combinatorial, primarily because he or sFis aware of certain standardC-,, -

tech-_ .-.---

niques for converting apparently non-combinatorial arguments into combina-

torial ones. Such subtleties will not concern us here, and we now give some clear-

tut examples of the distinction between combinatorial and non-combinatorial

proofs.

1.1.16 Example. Let n and k be fixed positive integers. How many sequences

(XI, X2.. . . , XJ are there of subsets of the set [n] = { 1,2,. . , nl such that XI n

Xl n ... n Xk = @? Let f(k, n) be this number. If we were not particuiarly in-spired we could perhaps argue as follows. Suppose XI n X2 n . . . n Xk_l = T,

wherelTl=~.If~=Xi-T,thenY~n...nY~_~=~and~~[n]-T.Hence

there are f(k - 1,n - i) sequences (XI,. . .,Xk_l) such that XI n X2 n ... n

Xk-l = T. For each such sequence, Xk cari be any of the 2”-i subsets of [n] - T.

As is probably familiar to most readers and will be discussed later, there a re

(y) = n /i (n - i) i-element subsets T of [n]. Hence

stanley ec

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