Combinatorial models for computations with formal power series

Table of contents :
Abstract 6
Declaration 7
Copyright 8
Acknowledgements 9
The Author 10
1 Introduction 12
2 Background and Definitions 17
2.1 P o sets ..................................................................................................... 17
2.2 Run Decomposition ............................................................................... 20
2.3 Avoidance Partitions and W irings ....................................................... 22
2.4 Combinatorial Correspondences ......................................................... 27
2.5 Infinite Lower Triangular Matrices...................................................... 40
2.6 The Landweber-Novikov A lgebra ....................................................... 49
3 Functional Inversion 51
3.1 Combinatorial Objects for Functional Reversion ............................ 53
3.2 Combinatorial Bijections ...................................................................... 64
4 Functional Composition 70
4.1 Combinatorial Objects for Functional Composition ........................ 71
4.2 Combinatorial Bijections ...................................................................... 77
5 Cell-Sets 79
5.1 Generalities on Cell-Sets ...................................................................... 79
5.2 The Cohomology of (CP1)1 1 ................................................................ 85
5.3 Bounded Flag Manifolds ...................................................................... 88
5.4 Combinatorial Models for the Coproduct ......................................... 97
5.5 Further Properties of the Posets Pn ...................... 102
6 Interval Coproduct 104
6.1 Interval Cell-Sets ...................................................................................... 104
6.2 The Lattice of Non-Crossing Partitions ................................................ 105
A Numbers and Formulae 114
A.l Catalan and Motzkin Numbers ............................................................. 114
A .2 Catalan Powers ............... 116
A.3 Narayana Numbers ................................................................................... 117
A.4 Fuss Numbers ............................................................................................. 119
Bibliography 122

Citation preview

COMBINATORIAL MODELS FOR COMPUTATIONS WITH FORMAL POWER SERIES

A THESIS SUBMITTED TO THE UNIVERSITY OF M ANCHESTER FOR THE DEGREE OF D O C TO R OF PHILOSOPHY in t h e F a c u l t y o f S c i e n c e a n d E n g i n e e r i n g

July 2000

By Andrew R. Maynard Department of Mathematics

ProQuest Number: 10833588

All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is d e p e n d e n t upon the quality of the copy subm itted. In the unlikely e v e n t that the a u thor did not send a c o m p le te m anuscript and there are missing pages, these will be noted. Also, if m aterial had to be rem oved, a n o te will ind ica te the deletion.

uest ProQuest 10833588 Published by ProQuest LLC(2018). C opyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

JO H N RYLANDS UNIVERSITY LIBRARY OF MANCHESTER

C on ten ts

A bstract

6

D eclaration

7

Copyright

8

Acknow ledgem ents

9

The A uthor

10

1 Introduction

12

2 Background and D efinitions

17

2.1

P o s e t s .....................................................................................................

17

2.2

Run D ecom position...............................................................................

20

2.3

Avoidance Partitions and W irin g s.......................................................

22

2.4

Combinatorial C orrespondences.........................................................

27

2.5

Infinite Lower Triangular M atrices......................................................

40

2.6

The Landweber-NovikovA l g e b r a .......................................................

49

3 Functional Inversion

51

3.1

Combinatorial Objects for Functional R ev e rsio n ............................

53

3.2

Combinatorial B ijectio n s......................................................................

64

2

4 Functional C om position

70

4.1

Combinatorial Objects for FunctionalComposition

........................

71

4.2

Combinatorial B ijectio n s......................................................................

77

5 C ell-Sets

79

5.1

Generalities on C e ll-S e ts......................................................................

79

5.2

The Cohomology of (CP1)1 1 ................................................................

85

5.3

Bounded Flag M an ifo ld s......................................................................

88

5.4

Combinatorial Models for the C o p ro d u c t.........................................

97

5.5

Further Properties of the Posets Pn ......................

6 Interval C oproduct

102

104

6.1

Interval C e ll-S e ts ...................................................................................... 104

6.2

The Lattice of Non-Crossing P a r titio n s ................................................ 105

A N um bers and Formulae

114 .............................................................114

A .l Catalan and Motzkin Numbers A .2 Catalan P o w e rs ...............

116

A.3 Narayana N u m b e rs................................................................................... 117 A.4 Fuss N u m b e rs.............................................................................................119 B ibliography

122

3

List o f F igures 2.1

An example of a multiarc and how it could be redrawn...................

2.2

The wiring corresponding to the partition 19/26/35/4/78 and the

23

sequence 1,2,3,4, 7,10,11,13,17 , an indication of how the dual is formed and its dual partition 1/249/3/58/67 and the sequence 1,2 ,3 ,6 ,7 ,8 ,1 0 ,1 3 ,1 6 ............................................................................. 2.3

24

The labels (below) of a wiring in relation to the labels (above) for the dual wiring.........................................................................................

25

2.4

Labelling of the squares which might be above or below a Dyck path. 28

2.5

A planted plane trivalent tree and the same tree with its trivalent vertices labelled with the first labelling and first/second labelling.

2.6

29

Two trees and the tree formed by merging their roots and inserting a planting leaf.

...................................................................................

35

2.7

The Dyck path which corresponds to the sequence 1245..................

35

2.8

Wirings using 3 2-arcs, a 2-arc and a 3-arc and finally using just one 4-arc................. ..................................................................................

2.9

Example of the algorithm from which a wiring is obtained from the chain 0 C {2} C {2,3} C [3]............................................................

3.1

46

47

An illustration of the bijection between polygonization of a polygon and multi-parentheses of a string of variables.....................................

4

64

3.2

An illustration of the bijection between polygonization of a polygon and planted plane trees...........................................................................

3.3

65

An illustration of the bijection between planted plane trees to neu­ tral chains. This tree gives the neutral chain 0 C {3} C {1,2,3} C {1, 2, 3, 4} ......................................................................................................

5.1

66

The eight planted planar trivalent trees with four trivalent vertices and of length five labelled by the first labelling algorithm together with their contribution to the coproduct for y[3]................................

88

5.2

Hasse diagrams of P 2 and P 3.................................................................

89

5.3

Hasse diagram of P 4................................................................................

91

5.4

Hasse diagram of the ordering < Q at n = 4........................................

94

5.5

An illustration of the equation x \x \x \ — X2 Xzx±x$x§x%x§ in X n for n > 9..........................................................................................................

5.6

96

The fourteen planted plane trivalent trees with four trivalent ver­ tices labelled using the first labelling algorithm together with their contributions to the coproduct for y[3]..................................................... 101

6.1

The lattice of non-crossing partitions AT7(4).........................................106

6.2

An example of non-isomorphic subintervals which both have type { 1, 2}..............................................................................................................107

A .l Combinatorial illustrations of the Catalan and Motzldn numbers.

5

115

A b stra ct In the first half of the thesis we give combinatorial descriptions of composition and reversion of special reversible power series, which are known to be implicitly encoded in the Landweber-Novikov algebra as the product and antipode. It is well known that the antipode of a Hopf algebra can be described in terms of chains in posets; indeed in [9] the antipode in the Landweber-Novikov algebra is described in terms of chains in Boolean posets. We classify chains in Boolean posets into three types and thereby obtain a new description of the antipode of the Landweber-Novikov algebra. In the second half of the thesis we review the notion of cell-sets [10] and discuss the dual homology coproduct of two different topological spaces in this framework. From this, combinatorial relationships between a functional inversion formula and the cohomology of the bounded flag manifolds are demonstrated.

6

D eclaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institution of learning.

7

C opyright Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instruc­ tions given by the Author and lodged in the John Rylands University Library of Manchester. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in the University of Manchester, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploita­ tion may take place is available from the Head of the Department of Mathematics.

A ck n ow led gem ents Firstly I would like to thank my supervisor, Nige Ray, for all the time, effort and inspiration he contributed to this research.

The financial assistance provided my parents, without which this research would not have been possible, is gratefully acknowledged.

Finally I would like to thank my girlfriend Collette for doing the washing up (properly).

This thesis is dedicated my parents: Jan and Pete.

9

T h e A u th or The author was awarded the degree of B.A. Mathematics (First Class) by the University of Cambridge in 1991 and then went on to obtain a pass in part three of the tripos. After this an MPhil in theoretical physics was offered by the University of Liverpool. This PhD was initiated in October 1996.

10

I have found a tru ly rem arkable proof which this m argin is not wide enough to contain. - Translation of a comment by Fermat written in the margin of his copy of Disquisitiones Arithmeticae.

11

C hapter 1 In trod u ction This thesis is concerned with aspects of algebraic combinatorics which are inspired by algebraic topology, in the hope that the combinatorial viewpoint might offer some alternative insights into certain spaces which arise in complex geometry. These spaces decompose naturally into cells (as CW complexes, as topologists call them), and their complex origins ensure that all the cells have even dimensions. In this situation, the integral homology and cohomology groups may be computed in terms of cellular chain complexes, and are free. We will not concern ourselves with many of the topological details here, which can easily be found in classic textbooks such as [8]. Our programme is inspired by the work of Gian-Carlo Rota and his school [5], who first suggested that many coalgebraic structures are inherently combinatorial, and could profitably be treated as such. We may consider a binary operation essentially as a rule for taking two objects and putting them together to create one object. A coproduct is dual, where we are given a single object and enquire as to all the ways of breaking it into two subobjects. The relationship with a space X whose cells lie in even dimensions arises because the cellular chain complex C*(X) is a coalgebra under the map induced by the diagonal X —>■ X x X .

Since the odd dimensional groups in (^ (X ), 12

CH APTER 1. INTRODUCTION

13

and therefore all its differentials, are zero, we may interpret this coalgebra as the integral homology

so that our calculations are more familiar. For

example, the coproduct is dual to the cup product in the integral cohomology ring H *(X)f and the duality is expressed by the universal coefficient theorem H 2k{ X ) ^ H o m ( H 2k{ X ) ^ ) , for each k ^ 0 . In this situation, we aim to introduce a combinatorial object, or class of such objects, to which we can associate the graded coalgebra H*(X) by some functorial procedure. Philosophically, the combinatorial setup may then be interpreted as an approximation to the geometrical object X , as explained in [10]. We concen­ trate here on the combinatorics and the algebra, investigating several important examples which we hope will eventually lead to further development of more categorical aspects of the theory. The combinatorial objects we shall deal with include chains in partially or­ dered sets, which can be broken into subchains, and planted trees, which can naturally be broken at their root to give a collection of smaller trees. It is well known th at planted plane trivalent trees (weighted by the number of vertices) enu­ merate to the Catalan numbers, and a considerable part of our work is involved in studying these numbers, and various related sequences. Other combinatorial objects will also be discussed, most notably integer sequences, Dyck paths and certain kinds of partition. The simplest example of a space where our programme can be followed is the complex projective space CP n. Here the cohomology ring is a truncated polynomial algebra H *{£Pn ,Z ) = Z[x]/{ £n+1 = 0 ); the dual homology coproduct H*{£Pn) -> H ,{£ P n) ® H ,{ £ P n)

CH APTER 1. INTRODUCTION

14

can then be described by

k = y ^ y j® y k - i\ i-o the associated combinatorial object is simply a straight line consisting of n points, and the coproduct formula above tells us all the ways of chopping up a line of length k into two sublines. This principle was formalised in [10] by introducing the category of cell-sets. A similar procedure may be followed for many other well known examples, such as Grassmannian manifolds and flag manifolds. We are particularly inter­ ested in the bounded flag manifolds B n, which were originally introduced by Bott and Samelson [1], and have more recently been studied in [2]. The latter work links the B n to the dual of the topologist’s Landweber-Novikov algebra; for our combinatorial purposes, it suffices to describe this as a Hopf algebra whose co­ product is represented by composition of formal power series, and whose antipode is represented by reversion. A flag is a sequence of nested subspaces of Cn 0 = VQ< V 1 < - - - {0, 1,. .. ,ra} and the map p is referred to as the grading. Given a graded poset one can associate a series known as the Poincare polynomial (when P is finite) or Poincare series (when P is infinite.) It is defined by Ylx2k2 —l ] U ' **U

, CLr + kT —1]

(^*^9

for i = 1 , . . . , r —1 and each [a*, a* + ki —1] is an interval in

the poset [n]. Expression (2.2) is the run decomposition of A . We shall also be interested in the length of the blocks and use the following notation for them bl(A) = (ku k2} • • ■,k r). The number r of blocks in .A will be referred to as the length of the run decom­ position and will be denoted by lrd{A). Let B be another element of the power set with run decomposition B

=

[a'u a[ + k[ — 1] U [a'2, a’2 + k'2 — 1] U • *• U [a's, a's + k!s — 1].

We refer to the interval[a,., ar + kr — 1] as the highest block of A and say it is above the highest block of B if ar > o!s + k's. The information of the block sizes of the run decomposition can be encoded algebraically in the polynomial algebra Z[&i, &2, • • •] on an infinite set of independent variables and we shall denote by r a type which can be associated to a pair of comparable elements of the power set (with the usual ordering). First we define that when the smaller of the two elements is the empty set then r is given by r

r(A) = r[0,^] = n 6'=. i=1

(2-3)

where ki is the length of the ith block of A. Now given B C [n] we can define a projection

ttb,

from the power set of [ra] to the power set of [ra] \ B, by ttb (A)

= A - A n B.

CHAPTER 2. BACKGROUND AND DEFINITIONS

21

An element A of the power set maps to the set which consists of those elements of A which are not in B. In particular 7Tb([7i]) is the complement B of B in [n], which is an (n—| B | ) —set in [n] and can be mapped into [n—|I?|] uniquely by requiring th at a map j : B —>■ [n—|i?|] preserves the order of the elements of B (ordered as a subposet of [n]) and those elements of [n—|i?|]. More explicitly, for aq, a2 G [n] — B cq < a2

j(ai) < j{a 2)\

j is a bijection and therefore has an inverse. Thus we have an isomorphism of posets B + = { i e 2 ^ | B C A C [n]} and

The map ttb ' B + —>■2^n~ ^

is the map which deletes B from each of the elements of B + and the remaining elements are relabelled by the map j. We denote by A# the map which is “inverse” to 7Tg. More explicitly, given an element of [n—\B\] relabel the elements by j " 1 then form the union with the set B (this union will necessarily be a disjoint union). Lem m a 2.2.1. I f B C A C [n] then 7Tb [B,A] = [0, 7Tb(A)] and if C C B then \ b [®,C] = [ B ,\ b (C)] and both

ttb

and Ab preserve order.

Proof. The interval [B, A] is by definition [B,A] = {C £[n] | B C C C A } . Now for B C Ci C C2 C A it follows that 0 C ttb{Ci) C 7rB(C2) C 7Tb(A) and so ttb preserves order and the image of the interval [B, A] under 7r# is the interval [0,ttb (A)]. The second part follows similarly.

□

CHAPTER 2. BACKGROUND AND DEFINITIONS

22

The type of a comparable pair of elements B C A is now given by t [ B , A ] = t (wb {A)).

2.3

A void an ce P artitio n s and W irin gs

A partition 7r of a set A is a collection of pairwise disjoint nonempty subsets £?i,. . . , Bf. of A whose union is A. A partition

tt of

A is said to be a refinement

of a partition a of A , if each of the blocks of a can be expressed as a union of a blocks of 7T. It is well known that the set of partitions form a poset with ordering being defined by refinement. We now let A = [n] and make the following definitions which will be useful later. A partition is said to be a delineation or 13/2 avoiding partition if given a,c E B i, b E B j and a < b < c then i — j. A partition is said to be non-crossing or non-nesting or 13/24 avoiding if given a, c E Bi, b,d E B j and a < b < c < d then i = j. Next we introduce some combinatorial objects which will be used later; we shall call them wirings. Before defining a wiring we shall define a multiarc. In the definitions that follow all the objects will be drawn in a two dimensional plane and everything will lie in the upper half of this plane. Place n points in the plane lying on a horizontal line (this horizontal line breaks the plane into the upper and lower half planes); then place a point p in the interior of the upper half plane and join it to a collection of the points on the horizontal line with straight line segments. The collection of segments from p to the points on the line is a multiarc. Often when depicting these objects we shall not use straight line segments but curves and also a given multiarc may apparently have several p points. Of course this does not affect the above definition and serves only to make drawings of multiarcs clearer (see Figure 2 .1). A multiarc which joins k

CHAPTER 2. BACKGROUND AND DEFINITIONS

23

points together will be called a k —arc.

1

2

3

4

n

1 2

3

4

Figure 2.1: An example of a multiarc and how it could be redrawn.

Now we are ready to define a wiring. Place n points in the plane lying on a horizontal line, a wiring is a collection of multiarcs such th at each of the points is used by exactly one multiarc and distinct multiarcs do not intersect one another. Examples of these objects are given in Figures 2.8 and 2.2. We shall also say that a multiarc is joinable to another multiarc if there exists an arc joining the two multiarcs which lies entirely above the points and does not intersect any other multiarc. It is easy is to set up a one-to-one correspondence between non-crossing par­ titions and wirings. Given a non-crossing partition B i , . . . , Bk of [ra] a wiring can be formed as follows; label n points lying on a horizontal line with the elements of [ra] from left to right with the usual order. Now for each set Bi join together those points which correspond to the elements of Bi with a multiarc above the points. It is easily seen th at multiarcs need not intersect one another by virtue of the requirement that the partition be 13/24 avoiding. An example is shown in Figure 2 .2 . Another set of combinatorial objects which will play a major role later is the set of integer sequences b± < ■■• < bn with i < bi < 2i —1 . These objects can be seen to be in one-to-one correspondence with wirings and non-crossing partitions as follows. From a wiring an integer sequence of the above type can be obtained as follows. Move from left to right across the points on the horizontal line, set a counter to 1 and record b\ — 1 for the first point and proceed inductively as

24

CH APTER 2. BACKGROUND AND DEFINITIONS

follows. Move from the ith to the (%+ l) tft point, increase the counter by one and if the point is joined to a previous point, the largest of which is j , then count the number of multiarcs which are to the left of the ith point and to the right of the j th point and to which this point is joinable, increase the counter by this amount plus one and record the value of the counter. If it is not joined to a previous point then leave the counter fixed and in either case record bi+i as the value of the counter. For example in the first diagram of Figure 2.2 as you go from the fifth to the sixth point the counter increases from seven to eight and there is one multiarc (namely the one th at joins the third and fifth points) so increase the counter by a further two to ten and record &6 = 10 but as you go from the eighth to the ninth point the counter increases from thirteen to fourteen and there are two multiarcs (namely the one joining two and six and the one joining seven and eight) so increase the counter by three to seventeen and record

= 17. This

process is easily seen to be reversible.

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

Figure 2.2: The wiring corresponding to the partition 19/26/35/4/78 and the sequence 1,2,3,4,7,10,11,13,17 , an indication of how the dual is formed and its dual partition 1/249/3/58/67 and the sequence 1,2,3, 6,7,8,10,13,16.

Next we define the notion of a dual wiring. The correspondence between wirings, non-crossing partitions and the integer sequences also give the definition of a dual partition and a dual sequence. First observe th at a wiring splits the upper half plane into a collection of path-connected regions (one of which is infinite). Now place new points on the horizontal line in between each of the existing points and place a point on the right hand end. Next join these together with a multiarc if they are in the same region. Finally reflect the newly created

25

CHAPTER 2 . BACKGROUND AND DEFINITIONS

wiring in a vertical line to give the dual wiring. This process is illustrated in Figure 2 .2 . The following Lemma will be crucial in setting up certain correspondences in Chapter three. L em m a 2.3.1. Given an integer sequences b\ < • • * < bn+1 with i < 6* < 2 i — 1 and the dual sequence c\ < • ■• < cn+i then bi = bi+i + 1 if and only if cn+i_j > Cn—i

1■

Proof. The proof of this Lemma relies on the correspondence between wirings and integer sequences of the type previously described. If b%= 6*+i + 1 then the ith point of the initial wiring is not joined to anything to the left and so in the dual wiring the n — i + 1 th point must be joined to something to the left and so cn- i +1 > cn-i —1 in the partition to which it corresponds (consider the diagram in Figure 2.3.) n -i+ 1

1

i-1

2

.1

n -i

i

i+ 1

n -1

n

Figure 2.3: The labels (below) of a wiring in relation to the labels (above) for the dual wiring.

The reverse implication also follows by a similar argument.

□ In the following Lemma we shall require the blocks of our non-crossing par­ titions to be ordered by the size of their smallest elements and thus 1 G B\. Let aj denote the smallest element in the ith block and c* denote the size of the ith block. Let a* = a[ — 1 and form the sets A = {a^ | i = 2 , . . . , k} and C = {bi | b{ =

cj

i — 1, ■■■,& — 1}. Now ai < bi, for if not then ai > bi

CH APTER 2. BACKGROUND AND DEFINITIONS

26

and the set [aj is larger in cardinality than [6J but the first i blocks contains bi elements and so there exists x £ [aj such that x £ Bj for j > i which contradicts the hypothesis that the blocks have been ordered by the size of their smallest elements. We therefore have Oj < bi for %= 1 , . . . , k — 1. Thus A and C have the same cardinality and satisfy A 0 for i = 0 , . . . , n — 1 and di+i < di + 1 can be obtained as follows. For a given Dyck path place it onto the template in Figure 2.4; then an integer sequence can be constructed by the following process: di takes the value j where j is the largest j such th at ij is a square below the Dyck path (if there are no squares under the Dyck path of the form ij then j is taken to be zero). We obtain a correspondence between planted plane trivalent trees and non­ crossing partitions as follows. Given a tree, label the trivalent vertices using the first labelling algorithm; now start at the trivalent vertex labelled with 1 and recursively form the set B\ as follows. To start, 1 £

and while a trivalent

vertex in B i is right occupied then include the label, into B \, of the trivalent vertex which right occupies it. Find the smallest label of a trivalent vertex which is npt in a block, then form a new block containing this label and use the rule while a trivalent vertex in Bi is right occupied then include the label, into Bi, of the trivalent vertex which right occupies it. Continue this procedure until all the labels have been put into blocks. A correspondence between non-crossing partitions of [n] and wirings of n points was described in Section 2.3. A correspondence between Dyck paths and sequences of l's and —l's is ob­ tained; given a Dyck path move from left to right and record a 1 if the Dyck path goes up and —1 if the Dyck path goes down.

CHAPTER 2. BACKGROUND AND DEFINITIONS

37

Finally a correspondence between wirings of 2n points using 2—arcs and se­ quences of l's and —l's is obtained as follows. Given a wiring move from left to right and record a 1 if the point is joined to a point to the right and —1 if the point is joined to a point to the left.

□ In [13] the Motzkin numbers are defined by their generating function which is ( . m ix)

=

v-—v. ; 1—x —v^l —2x —3x 2 > m ix-= ------------------------------4i=Q ^ 2 x2

=

1 + x 4 - 2x 2 4- 4a:3 + 9a;4 + 21a;5 -f- 51a;6 + • • •

We note some combinatorial objects which enumerate to these coefficients. Lem m a 2.4.3. The following are in one-to-one correspondence and enumerate to the Motzkin number m n: 1. Ternary words a i , a 2, . . . , a n such that the partial sums 5j[=i ai ^ 3 f or j ~ 1 , . . . , n - 1 and

ai = n -

2. Paths from (0,0) to (n, 0), with steps (1,1), (1 ,-1 ) and (1,0) never falling below the x-axis. 3. Wirings of n points using 1 -arcs and 2-arcs. 4 • Wirings of n p 1 points such that no two adjacent points are joined. 5. 123 and 13/24 avoiding partitions of[n\. 6.

13/24 avoiding partitions of[n + 1] such that no block contains consecutive elements.

Again before embarking upon the proof of 2.4.3 we illustrate it for n = 4.

38

CH APTER 2. BACKGROUND AND DEFINITIONS

1. Ternary sequences such th at the partial sums Yjl=i ai — 3 ^or 3 = 1>2»3 and 2 t=i ai = 41111

0112

0121

0211

1012

1021

1102

0022

0202

2 . Paths from (0,0) to (4,0), with steps ( 1 , 1), (1 , —1) and (1,0) never falling

below the below the a;-axis. /\/\

3. Wirings of n points using 1-arcs and 2-arcs.

f i n i n i 11n m i I r n Cn) n n

mm

Co)

4. Wirings of 5 points such that no two adjacent points joined.

mu ifni nm mu 11 fm fmi fmi 5. 123 and 13/24 avoiding partitions of [4]. 1 /2 /3 /4

12/3/4 1/23/4 1/2/34 13/2/4

1/24/3

14/2/3

12/34

14/23

6 . 13/24 avoiding partitions of [5] such that no block contains consecutive

elements. 1 /2 /3 /4 /5

13/2/4/5

1/24/3/5

1/2/35/4 14/2/3/5

1/25/3/4

15/2/3/4

15/24/3

135/2/4

39

CHAPTER 2. BACKGROUND AND DEFINITIONS

Proof. First we shall show th at the number of ternary words of length to which satisfy the criteria and contains the number 2 k times is ( n — 2k \

H

2fc

J

where c& is the kth Catalan number. The generating function can then be recov­ ered by elementary algebra. The proof will then conclude with bijections between the different objects. consisting of k 0rs and k 2rs and its partial

Consider the words &i,. . . , sums satisfying

^ J f°r

3

~ 1, ■■•, 2 & — 1 and

^ = 2 ^- If is easy

to see th at there are c&such words. Now given such a word insert (to —2k) l's between the entries of the word. It is easy to see that for each such word there are (” 2**) different ways to insert the l's and so the first part is shown. The correspondence between the ternary words and paths is given as follows. Given the ternary word ai, a 2>■■■?an if o>i = 0 then take the first step to be (1,1) and if cz,1 — 1 then take the first step to be ( 1, 0 ). More generally the ith step is formed from ai by the rule: if a; = 0 then take the ith step to be ( 1, 1), if a* = 1 then take the ith step to be (1 , 0) and if ai — 2 then take the ith step to be ( 1, —1). In Section 2.3 a correspondence between non-crossing partitions and wirings was described. Exactly the same algorithm gives a bijection between wirings using 1 —arcs and 2—arcs and non-crossing partitions which also avoid blocks of size three. This algorithm also gives a bijection between wirings using any multi­ arcs with no two contiguous points joined and non-crossing partitions with no consecutive elements in the same block. A correspondence between wirings using any multiarcs with no two adjacent points joined and ternary words such that the partial sums XXu j = 1 , . . . , 2 k —1 and

^ j f°r

^ ~ ^ Is obtained as follows; move from left to right

along the points and, if the first point is joined to another point on the right, then record a 2; otherwise record a 1 for the last letter of the word. More generally, if

40

CHAPTER 2. BACKGROUND AND DEFINITIONS

the ith point is joined to a point to the right, then record 2 for the (n + 1 —i)th letter of the word; if the (? + l ) th point is joined to a point to the left, then record 0 for the (n + 1 —i)th letter of the word; otherwise record 1 for the (n + 1 —i)th

letter of the word.

□ 2.5

Infinite Lower Triangular M atrices

Following [9] we construct an infinite matrix which displays in each row combi­ natorial information about the power set 2 ^ . This m atrix T& is in the group of lower triangular matrices over a polynomial algebra on an infinite set of indepen­ dent variables 2 LT(Z[b]). The (m, n)th entry is the sum

Tbl^i

types of (m —n )— sets in [m — 1]. The upper left corner is

Tb =

/ 1

0

0

0

h

1

0

0

2bi

1

0

\ (2.5)

bs 2&2 T b\ 3&i 1

/ Next we define a second matrix Tb.c in LT(Z[b,c]). The (m ,n )th entry is now given by the following sum

Aes™N 171~ 71 B^ A 2We shall often abbreviate an infinite set of variables &i, &2 j • • • to just &. It should be clear from the context that an infinite set is understood.

41

CHAPTER 2. BACKGROUND AND DEFINITIONS The reader can verify that the first few entries of the upper left corner are /

1

0

bi H- C i

1

&2 T 2 b\C\ + C2

£>3 + 2 &2 Ci 4-

V

0

4-

C3

&i + 2 6 2

4-

6&1C1 +

;

•••

O O * ' *

2(&i + Ci)

61C1 + 3&1C2

0

+

2c 2

3 (& i

1

0

+ C i)

1

;

• ■■

( 2 .6 )

T h e o re m 2.5.1. The matrix Tb.c is the product TbTc in LT(7i[b, c]). Proof. The (m, n )th entry in TbTc is given by

£

E

n tx w n

n C {3 } C { 1 ,3 } C { 1 ,2 ,3 } ;

both give contributions of

but with differing signs.

So now th at all the cancellations in (2.8) have been documented we shall focus our attention on neutral chains. First we give a Lemma which allows us to describe them more easily and then we shall state a Theorem which will give another combinatorial description of these objects in terms of wirings. Lemma 2.5.4. A comparison B C A in 2^ is neutral if and only if

ttb{A)

=

[a, a + k — 1] (for some a^k > 0) and the highest block of A is either above or intersects with the highest block of B .

46

CHAPTER 2. BACKGROUND AND DEFINITIONS

nnn

nfnl

nm

1n 11

HTln 11n 1

1n n 1 IIFTI mn

Im 1

rm Figure 2.8: Wirings using 3 2-arcs, a 2-arc and a 3-arc and finally using just one 4-arc.

This Lemma follows quickly from the definitions and interested readers can easily furnish the proof for themselves. Given a chain in 2 ^ 7 : 0 = A0 C A i C • • • C Ai = A let Si denote the cardinalities |

—Ai | for i = 0 , . . . , I —1. In the next Theorem

a bijection between chains of the above type and wirings of I + X ^ = o p o i n t s using an (sj + l)-arc for i = 0 , . . . , I —1 will be shown. In the statement of the next Theorem these details concerning the sizes of the relative cardinalities between the comparisons in the chain and the multiplicities of the multiarcs in the wirings will be suppressed. Theorem 2.5.5, Neutral chains of [n] are in one-to-one correspondence with wirings. As usual we first illustrate this Theorem by a small example (n — 3). The five neutral chains in 2® with three relative cardinalities of one. which

47

CHAPTER 2. BACKGROUND AND DEFINITIONS correspond to the five wirings using three 2-arcs, are 0 c {l}< = {l,2 }c{l,2 ,3 } 0 C {1}C {1,3}C {1,2,3} 0 C {2} C {1, 2} C {1, 2,3} 0 C {2} C {2,3} C {1,2,3} 0 C {3} C {2,3} C {1,2,3}.

The five neutral chains in 2 ^ with relative cardinalities of one and two, which correspond to the five wirings using a 2-arc and a 3-arc, are 0 C {1} C {1,2,3}

0 C {2} C {1,2,3} 0 C {3} C {1,2,3} 0 C {1,2} C {1,2,3} 0 C {2,3} C {1,2,3}. The neutral chain in 2® with a relative cardinality of three, which corresponds to the wiring using a 4-arc, is 0 C {1,2,3}.

• 1

. n i i . 2

3

4

5

6

1 2

3

4

5

*n n * 1 2

3

1n

n

.

4

Figure 2.9: Example of the algorithm from which a wiring is obtained from the chain 0 C { 2 } C {2, 3} C [3].

Proof, (of Theorem 2.5.5) Given a neutral chain

CHAPTER 2 . BACKGROUND AND DEFINITIONS let nii =\ Ai — A*_i | for i = 1, . . . , I and the partial sums rij = X/J=i

48 anc^

n = n\. Place I + n points in a line and label them 1, 2 , . . . , I + n left to right. By hypothesis A \ — [a, 6]. Join together the points a, a + 1 , . . . , 6 + 1 with a multiarc. Now the recursive step is as follows: relabel the points 1, 2 , . . . , / + ni-i left to right ignoring those points which are part of a multiarc which has already been labelled. Next take those points of Ai —A ^ i and the next largest point which is not already joined to a multiarc and join these together with a multiarc. This process is illustrated in Figure 2.9. This procedure is reversed as follows. Given a wiring, label the points left to right with 1, ., , , n ignoring the last point of each multiarc. Next go to the rightmost multiarc and record the labels in the set B i which are at the endpoints of that multiarc and delete this multi arc. Now the recursive step is to go to the rightmost multiarc and record the labels in the set Bi which are at the end of that multiarc and now delete this multiarc. The chain is then reconstructed from the sets Bi for i = 1 , . . . , / by A j = B i U 3 , - ! U ■• • U

j = 1,..., I

Then the chain is given by 0 C A i C • • • C Ai = [n].

□ Other combinatorial objects which enumerate to coefficients in an inversion formula will be given in the next Chapter. The subsequent Chapter gives combi­ natorial objects which give coefficients which occur in the composition for formal power series.

CHAPTER 2. BACKGROUND AND DEFINITIONS

2.6

49

T he L andw eber-N ovikov A lgeb ra

The Landweber-Novikov algebra was introduced into algebraic topology around the 1960’s independently by both Landweber and Novikov. It is a Hopf algebra (a bialgebra with an antipode) which we shall describe more fully in Section 5.1. Here we shall actually describe the dual of the Landweber-Novikov algebra S*, as this has certain combinatorial descriptions which are of interest to us; first we describe the elements and then action of the five structure maps of S*. It is a graded polynomial algebra on an infinite set of generators

• • • over the

integers. Each generator bi is in grading i and by convention we require bo = 1, Multiplication }i : S’* S* — >S'* is the standard polynomial multiplication given on monomials by

The counit map e : S* — >Z takes bo to 1 and all other generators to zero. The coproduct A : S* — >S* (g> S* is given by the following formula

* (* » )= £ f o * ) i+j=k \h>0

) i

where the inner sum is raised to the power of j + 1 and then those terms in the ith grading are taken. The unit 77 : Z

S* maps 1 to 60- The antipode x : S* — * S*

is compatible with these four structure maps if the following diagram commutes.

s*

Z

s*

A s ,® s ,

— > s«® s.

We shall next make explicit the structure of this algebra for small degrees.

CHAPTER 2 . BACKGROUND AND DEFINITIONS

50

From the coproduct formula one readily obtains ^ (^ l) =

61 ® 1 “h 1 ® &i

A (62) “ ^ ( 63) =

® bi p 1 ® b2

&2 ® 1 d-

^3 ® 1 + (262 P b\) ® b\ P 3&i 0 62 T 1 & ^3-

Using these and the requirement th at the above diagram commutes we derive %(6i) = X { b 2) -

-b i 2 b\

- b2

x(^) — —55f + 5bib2 —63The reader will notice that the coproduct is very similar to the first column of entries in the matrix (2 .6) and the antipode is precisely the firstcolumn of entries of

(2.9).This is a consequence of the fact that the coproduct map encodes

composition of formal power series and the antipode encodesreversion. It may not be obvious from the above formulae that the antipode is an involution but when it is realised th at these are the coefficients of a power series for a functional inverse then it is easily seen.

C hapter 3 F unctional Inversion This Chapter describes 15 different classes of combinatorial objects which enu­ merate the coefficients of a certain functional inversion formula. The Lagrange inversion formula [13] (superscripts mean “to the power of” and subscripts mean “extract the terms of th at degree”)

gives a relation between the coefficients of a formal power series f ( x) € a;lX[[x]] and the coefficients of its functional inverse f^~^(x) (i.e.

— x). Here

K[[a;]] denotes the ring of formal power series over some commuting field K with a unit. A simple example is provided by the function f ( x) = x + a ix2 and it is a routine calculation to show oo / [“ 1](z) = ^ ( - l ) ncnaia;n+1 71=0

where cn are the Catalan numbers. We shall perform this calculation for the more general series oo

f ( x) = x + T ; ajXt+l »=i 51

52

CHAPTER 3. FUNCTIONAL INVERSION where we have explicitly up to 5th order f ^ \ x ) = x — a \x 2 + (2a^ —a^)#3 + (—5a3 + 5a ia 2 —as )#4 + -|-(14a4 —21a^a2 4- 6aia 3 + 3a 2 —a^)x5 + • ■• .

(3.1)

More generally with a simple inductive argument 3.1 can be rewritten as = E ( - ^ C n u n ^ n ^ a T ■■•C ^X 1^ . where the coefficients are positive natural numbers and n =

(3.2) *ni an(^ Wl —

]C?=i n i an 0 and {/31}. . . , j3mt} is the same multiset as { l712,

—l) nfc}.

((0, 0), (0, 0))

((1, 0), (1, 0))

((2, 0), (2, 0))

((0 , 1), (0 , 1))

((1, 1), (1, 1))

((0 , 0), (1, 0))

((1, 0), (2, 0))

((0, 1), (1, 1))

((0, 0), (0 , 1))

((1, 0), (1, 1))

((0, 0), (1, 1)) {(0,0,0),

0,0,0))

( ( 1 , 0 , 0 ) , ( 1 , 0 , 0) )

( ( 2, 0, 0) , (2, 0, 0) )

( ( 3 , 0 , 0 ) , ( 3, 0 , 0 ) )

((0,1,0), (0,1,0))

{(1.1,0),

1,1,0))

( ( 2, 1 , 0 ) , ( 2, 1 , 0 ) )

( ( 0, 2, 0) , (0, 2, 0) )

((1,2,0), (1,2,0))

( ( 0 , 0 , 1), ( 0 , 0 , 1 ) )

((1,0,1),

1,0,1))

( ( 2, 0 , 1 ) , ( 2, 0 , 1 ) )

((0,1,1), (0,1,1))

((1,1,1), (1,1,1))

-

((0,0,0),

1,0,0))

( ( 1 , 0 , 0 ) , ( 2 , 0 , 0) )

( ( 2, 0, 0) , ( 3, 0, 0 ) )

((0,1,0), (1,1,0))

((1,1,0), (2,1,0))

((0,2,0),

1,2,0))

((0,0,1),(1,0,1))

( ( 1 , 0 , 1 ) , ( 2, 0 , 1 ) )

((0,1,1), (1,1,1))

((0,0,0), (0,1,0))

( ( 1 , 0 , 0) ,

1,1.0))

( ( 2, 0 , 0 ) , (2, 1 , 0 ) )

((0,1,0), (0,2,0))

((1,1,0), (1,2,0))

( ( 0 , 0 , 1) , ( 0 , 1 , 1 ) )

((1,0,1),

1,1,1))

((0,0,0), (0,0,1))

((1,0,0), (1,0,1))

((2,0,0), (2,0,1))

((0,1,0), (0,1,1))

((0,0,0),

1,1,0))

( ( 1 , 0 , 0 ) , ( 2, 1 , 0 ) )

((0,1,0), (1,2,0))

((0,1,1),

1,2,1))

((0,0,0), (0,1,1))

( ( 1 , 0 , 0) , ( 1 , 1 , 1 ) )

o o o

((1,1,0), (1,1,1))

1,0,1))

( ( 1 , 0 , 0 ) , ( 2, 0 , 1 ) )

((0,1,0), (1,1,1))

( ( 0 , 0 , 0), ( 1 , 1 , 1 ) )

CHAPTER 3 . FUNCTIONAL INVERSION

63

15. Pairs of integer sequences ai < a • • • 1

bn) | bn—i ~

a i+2

a i+ l

1

i=

0 , . . . , 72.

1},

with the ordering given by (di, d2, . . . , dn) > (61, &2>• • ■, K ) if and on-ly if

> h

for each i. Examples of these posets when n is equal to two, three and four can be found in Figures 5.2 and 5.3. We shall also take this opportunity to introduce the reflexive antisymmetric relation >:, defined on the set Pn) given by (di, d2, . . . , dn) h (&i, &2, ■. •»bn) if and only if di = h+ei for each i and e$ G {0,1}. It is clear th at > is the transitive closure of K We shall now state some elementary results about the algebra X n. L e m m a 5.3.1.

1. In X n the relation X^ — X^X^i • • •

holds for i + k < n + 1.

90

CHAPTER 5. CELL-SETS

2. Given a sequence of integers ai, • • • o>k there corresponds the monomial x f 1 • • • a in X n, define the partial sums fij ~

anV ° f these partial sums

satisfies (3j > n —j + 1 then the monomial is zero; in particular any term in X n of degree higher than n is zero. Proof, x f = XiX^+l follows by induction on k and use of the relation xj = XjXj+ Now using this and induction again X?

=

X ix'l'l = XiXi+1X ^

XiXi+i

= ... =

• •■ Xi+k-l.

To see the second part, consider the element (5.9) By hypothesis we can choose j such that (3j = Yli=j a i > n — j + 1. So by 1 ■ *— - / Y ^ l

4,01

'“ k

^ rp . , . . rp .

, , *

1

j —1

3

_

. »

J + P j“

but j + fij — 1 > n and x n+i = 0 and so the term in equation 5.9 is zero.

□

Let aH)k denote the number of elements of Pn such th at the sum of the coor­ dinates is k\ in other words, an^ is the cardinality of the set n

Pn,k = {{bi, • • •, bn) 6 Pn I ^

bi = k}.

(5.10)

i= 1

A tabulation of these numbers can be found in an Appendix to this thesis where they are called Catalan powers. W ith these conventions we are ready to state the following result. L em m a 5.3.2. The term x" in X n is equal to t Q1r a'2

7-a«

for each (ai, 0*2, . . . , ajn) 6 Pn)U and the cardinality of Pn^n is cn (the nth Catalan number).

CH APTE RS.

91

CELL-SETS

Figure 5.3: Hasse diagram of P 4

Proof. We shall begin by considering monomials cc"1 ■• ■x®n of degree n and show that these correspond to the elements of Pn_i. It is then easy to see that Pn}U has the required cardinality by establishing a one-to-one correspondence between Pn^n and Pn-\.

The proof is then concluded by showing that the monomials

ft*1 ■• • ftjn, for each (a 1} 0:2, . . . , ctn) E Pn,n» are equal. Given a sequence aq, 0:2, • •. oin of integers such that denote the partial sum YH-j

a* = n, let f t- j+ i

^or 0 ^ 3 5= 71• ^ i-s clear from part 2 of Lemma

5.3.1 th at if Pn~j+i > n — j + 1 then the element ft^ft^2 • • ■

will be zero. So

we have 0 < f t < j.

Now let ai = 1 and ai = /?n_i + i for %— 2, . . . , n. This integer sequence of a ’s satisfies a\
} which will be a non-crossing partition which contains no block with consecutive elements. This map between non-crossing partitions and non-crossing partitions with no block containing consecutive elements is many to one. Now by Lemma 2.4.3 we know th at such a combinatorial class of objects is enumerated by the Motzkin numbers and again by Lemma 2.4.3 is in correspondence with ternary words V\^ • • ,v n such th at the partial sums /3m =

v* for m = 1 , . . . , n satisfy

< n 4-1 —m

with equality when m — 1. We shall see that the ternary word given to a non­ crossing partition by the above procedure and the associated ternary word are the same. First we shall show that the process of splitting a block at an occurrence of consecutive elements does not effect the associated ternary word. Given

tx

=

{ 5 i , . . . , Bk} and a block with consecutive elements p ,p P 1 G B q. By Lemma 6.2.1 'T'lpArcCn+i); 7r] = B[ U ■■■U B'q U • • • U B'k where B^ = Bi — {^} and U is the largest element of Bi and

t [tx,

ljvc(n+i)] = U-L2{si —1} where s* is the smallest element

of Bi. Now break the qth block into two new blocks B^~ — {x and B ^ = {x

E

E

B q | x < p}

B q \ x > p + 1} to obtain 7r*. We have t[0wc(th-i)> tx*] =

B[ U • • • U B ^ ' U B ^ ' U • • • U B'k+1 and r[lwc(n-H), n*] =

~ 1}- So now

p §£ ro, but p + 1 is the smallest element of the block B ^ ' and so p G n - So breaking a block has the effect of moving p from To to T\ and so wp = 1 and the rest of the ternary word is unaffected. Finally we shall show that the ternary word obtained from the non-crossing

CHAPTER 6. INTERVAL COPRODUCT

110

partition with no block containing consecutive elements gives the same ternary word as the associated ternary word. In Lemma 2.4.3 the correspondence between non-crossing partition with no block containing consecutive elements and wirings using multiarcs such th at no two adjacent points are joined was shown and then a correspondence between wirings using multiarcs such that no two adjacent points are joined and ternary words «!,••• , vn such that the partial sums /3m =

v*

(m — 1 , . . . , n) satisfy {3m < n + 1 —m with equality when m = 1 was shown. The correspondence between non-crossing partitions with no block containing consecutive elements and ternary words can be described more directly as follows. Given a non-crossing partition

tt =

{B i , . . . , B^} with no consecutive elements in

the same block let p and p + 1 be in the qth and q th blocks; then if there exists p' G B q such th at p < p then wp — 2; if there exists p1 G B q>such that p + 1 > p then wp = 0; and if neither of the above then wp = 1. Now for a non-crossing partition 7r = {Lh,. . . , B &} with no consecutive elements in the same block, a G r0 if a G Bj for some j , so there exists a G Bj and a + lG Bj> for some j' and this is the smallest element in this block (otherwise it would contradict the partition being 13/24 avoiding), so To C n and wa = 2 which is equivalent to the above criteria. If p + 1 G B j for some j and p

ti then p + 1 is not the smallest element

in the block B j; thus the criteria for wp = 0 are seen to be same and the Lemma is proved.

L em m a 6.2.3. Let E(w) denote all those non-crossing partitions of N C (n + 1) which have the ternary word w and let Ai(w) denote all those entries ofw whose value is 1; then the cardinality of E(w) is 2 ^ —^, and for 7Ti, 7r2 £ E(w) and 7Ti ^ 7r2 then

^ Tj(7t2) for i — 0,1.

Proof. Let 7Ti be a non-crossing partition whose associated ternary word is w. As in the previous Lemma we can go through the blocks of 7Ti, and those blocks

CHAPTER 6. INTERVAL COPRODUCT

111

which contain consecutive elements can be replaced by two blocks which have been broken at the point of the two consecutive elements. It was also shown that if p and p + 1 where in the same block of 7Ti then p

E

To and p ^[r\\ when this

block is refined into two blocks then p is moved from r 0 to p

and so p ^ r0 and

ri. let the associated binary word of the non-crossing partitions

G

tti

be w and

let j4i(w) = {i | Wi = 1}. There are two possiblities for each of the elements of p

E

Ai(u;) either p

E tq

or p

E t\

and there are

possible non-crossing

partitions with w as their associated binary word. It is easy to see th at if we have two non-crossing partitions with the same associated ternary word then for %E Ai(w) either i

E tq

or i

E

Ti depending on

whether i and %+ 1 are in the same or different blocks.

□ We shall give a small example to illustrate the lemma above; the partition 1 4 /2 3 has r 0 = { 1 ,2 } , n = {1 } and the associated ternary word is 210 now

break the second block and we have the the partition 1 4 /2 /3 which has To = {1 }, T\ = {1,2} and the associated ternary word is 210 and these are the two non­ crossing partitions with ternary word 210. Let J\fC(n-\-1) denote the set consisting of the lattice of non-crossing partitions N C (n -f-1) and all its subintervals. Next define a pair of intervals [tti, ) will give

different contributions to (6.1). Thus the correspondence between the terms in the coproduct of the longest word described in Lemma 5.4.2 and equation (6.1) is shown.

□ We shall conclude by mentioning another way to obtain a correspondence between the terms in the coproduct of the bounded flag manifolds and the chains of length 2 in the interval of non-crossing partitions. In Lemma 2.3.2 it was

CHAPTER 6. INTERVAL COPRODUCT

113

shown th at a non-crossing partition ir of N C (n + 1) corresponds to a pair of subsets (A,r,

o f [n] which are of equal cardinality and which satisfy

134/2] - { 1 ,2 }

t-1134/2, 1NC] = {1 } f [ 1 3 4 /2 ,1NC] = { 1 } .

N ext recall from Section 5.4 th at the coproduct of the longest elem ent has the following description

a

(?/m ) = Y2yu®yv

where the sum is on pairs of sets U and V which satisfy U C [n], V < a U and

U + V = [n]. The correspondence between the term s o f the above description and those given by equation (6.1) w ith r replaced by

f is easily seen when the non­

crossing partitions are described by pairs of subsets of [n] of equal cardinality and com parable w ith respect to < a . Thus we rewrite equation (6.1) as

A (?/[n ])

"y ^

ireNC(n+l)

2/f[0]VC)(j4ir).B7r)] ® ^[(A n-jBjr^liV C r]

A p p en d ix A N u m b ers and Form ulae This appendix collects together certain sequences and arrays of numbers which frequently occur in this thesis. We place them here in order to try not to break the flow of the main text and in order to note some of their more interesting properties.

A .l

C atalan and M otzk in N u m b ers

There are many ways [12] to define the Catalan and Motzkin numbers. Here we shall define them in terms of walks on the natural numbers and collect together some of their elementary properties. The Catalan number cn is the number of random walks on N, of length 2n, that start and end at 0, with the possible moves ±1. Similarly the Motzkin number m n is the number of random walks on N, of length n, with possible moves {—1,0,1} again starting and ending at 0. Let c(x) denote the generating function for the Catalan numbers and similarly m (x) for

114

APPEND IX A. NUMBERS AND FORMULAE

115

the Motzkin numbers; they satisfy the following formulae Cm. —

1

(2 n

c{x)

2x c(x) = 1 -f xc2(x)

m{x) m

1 —x — \ / l — 2x — 3x2 2x2

w = r b c( (1 ? r—^x )‘‘) -

The first few Catalan numbers are 1,1, 2,5,14,42,132,429,1430,... and the first few Motzkin numbers are 1,1,2,4,9,21,51,127,323,.... / \

/ \ / \ / \ /

\

/ \ / \

Figure A.l: Combinatorial illustrations of the Catalan and Motzkin numbers.

APPEND IX A. NUMBERS AND FORMULAE

A .2

116

C atalan Pow ers l

l

1

1

1

1

1

1

1

9

1

l

10

35

44

54

5

9

154

208

28

48

75

110

429

637

5

14

14

20

27

2

3

5

7

8

2

4

6

l

14

42

90

132

165

275

297

572

1001

1638

42

132

429

1001

2002

3640

429

1430

3432

7072

1430

4862

11934

4862

16796 16796

The numbers given above are referred to several times in the main text. In this appendix we note the recurrence relations that define these numbers (qj) and also the generating function for these numbers (0(a;, y) =

ci,3 x%y^)'

APPEND IX A. NUMBERS AND FORMULAE

117

They satisfy the following formulae ci,j —