Random Walks, Trees and Extensions of Riordan Group Techniques

Table of contents :
D issertation A pproval S h e e t ............................................................................
D e d ic a tio n .................................................................................................................
A c k n o w le d g m e n ts ................................................................................................
A b s tr a c t ....................................................................................................................
L ist o f F ig u r e s .......................................................................................................
C hapter 1. In tro d u c tio n ...................................................................................
1.1 Basic Definitions and O ve rvie w ..................................................................
1.1.1 Definitions .............................................................................................
1.1.2 Overview .............................................................................................
1.2 The Catalan Numbers: Some Interpretations and Related Results . .
1.2.1 Combinatorial Interpretations of the Catalan Numbers . . . .
1.2.2 Statistics Related to the Catalan N um bers..................................
1.2.3 Functions Related to C { z ) ...............................................................
1.3 The Ternary numbers and Generalized Dyck P a th s ..............................
1.3.1 Combinatorial Interpretations of the Ternary Numbers . . . .
1.3.2 Generalized i-Dyck paths ...................................................................
viii
ii
iii
iv
vi
x
1
1
1
6
8
8
10
10
12
12
17
1.4 The Riordan G roup ............................................................................................. 17
C hapter 2. Analogues o f Catalan P ro p e rtie s ............................................ 24
2.1 T(z) is to N(z) as C(z) is to B (z ) ................................................................... 24
2.2 The Chung-Feller Theorem for f-Dyck p a th s .............................................. 26
2.3 The Motzkin Analogue ...................................................................................... 29
2.3.1 The Balls on the Lawn P roblem .......................................................... 30
2.3.2 The Euler Transform .............................................................................. 33
2.4 The Fine A nalogue ............................................................................................. 35
C hapter 3. R eturns and A re a .............................. 38
3.1 Expected Number of R e tu rn s ......................................................................... 38
3.1.1 Dyck P a th s ............................................................................................... 39
3.1.2 Ternary P aths ........................................................................................... 39
3.2 Techniques for Finding Area Under Paths ................................................... 43
3.2.1 Dyck P a th s ............................................................................................... 44
3.2.2 Ternary P aths ........................................................................................... 48
C hapter 4. Conclusions and Open Q u e s tio n s ............................................ 53
4.1 Summary and Work to be D o n e ...................................................................... 53
4.2 More Open Questions .......................................................................................... 54
4.2.1 Elements of Pseudo Order 2 ................................................................. 54
4.2.2 Determinant Sequences ........................................................................... 57
4.2.3 Narayana Analogue .................................................................................. 62

Citation preview

INFORMATION TO USERS

This manuscript has been reproduced from the microfilm master.

UMI films

the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer.

The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction.

In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted.

Also, if unauthorized

copyright material had to be removed, a note will indicate the deletion.

Oversize materials (e.g.,

maps, drawings,

charts) are

reproduced

by

sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps.

ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

H O W A R D U N IV E R S IT Y R a n d o m W a lk s , T re e s a n d E x te n s io n s of R io r d a n G r o u p T e c h n iq u e s

A Dissertation S ubm itted to the Faculty o f the G raduate School

of

H O W A R D U N IV E R S IT Y

in p a rtia l fu lfillm e n t o f th e requirements for the degree

D O C T O R O F P H IL O S O P H Y

D epartm ent o f M athem atics

by

N a io m i T u e re C a m e ro n

W ashington, D.C. M ay 2002

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

UMI Number: 3066485

UMI UMI Microform 3066485 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

H O W A R D U N IV E R S IT Y G R AD U ATE SCHOOL D E P A R T M E N T O F M A T H E M A T IC S D IS S E R TA TIO N C O M M IT T E E

N eil H indm an, Ph.D. Chairperson

Paul Peart, Ph.D.

T jm oi % { j^ x L p iy -u ^ Louis W . Shapiro, Ph.D .

W en-Jin Woan, Ph.D.

^ o h W o o d s o n , Ph.D Associate Professor o f M athem atics Morgan State U niversity

Louis W . Shapiro, Ph.D . D issertation A dvisor Candidate: N aiom i Tuere Cameron Date o f Defense: January 17, 2002

ii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

D e d ic a tio n

Dedicated to m y beloved brother, Jarre ll A li Cameron (December 5, 1977 - September 14, 2001) You w ill live forever in m y heart.

iii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

A c k n o w le d g m e n ts

W h ile it is not possible to name every person who supported me during m y graduate studies, I w ould like to expressly th a n k those in dividuals whose support proved c ritic a l to the achievement o f m y goals. F irs t and forem ost, I th a n k m y Lord and Saviour, Jesus C h rist, through whom a ll things are possible. I would lik e to express m y most sincerest appreciation to m y beloved fam ily, Jason, K a riy m a and O m ar, for th e ir unconditional love and support throughout th is chapter o f our life. I could not have done th is w ith o u t th e ir support. I also want to th a n k m y M om , m y Dad, m y sister N je ri and m y b rother Jawanza for th e ir consistent love and understanding. I m ust also thank m y m other-in-law P atricia M u rp h y for her constant encouragement. M y advisor, Louis Shapiro, deserves the credit fo r raising m any o f the questions addressed in th is research, and I would like to th a n k h im fo r his care and guidance thro ugho ut th is proje ct. I m ust also thank the members o f m y com m ittee and the C om binatorics Seminar, Paul Peart, W en-Jin Woan, Seyoum G etu, Leon Woodson, N eil H indm an and E m eric Deutsch, for th e ir invaluable in p u t.

In addition, the

members o f the M athem atics Departm ent at Howard U n ive rsity have provided im ­ mense support and encouragement. In p a rticu la r, I would lik e to m ention Adeniran Adeboye, R ichard Bourgin, Francois Ramaroson, the graduate student body and the a d m in istra tive support staff. I m ust extend a special thanks to Cora Sadosky for her support and m entorship to me as a m athem atician and as a woman.

iv

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

There are m any extended fa m ily members and friends whose support of me and m y fa m ily have been crucial to the com pletion o f m y dissertation. They include Lisa and R ick Square, Karen McFadden, Carla Jones-Chew, Sherry Scott, L a tifa B a rn e tt, Deangela H ill, Rena Stevens, T ia ndra Speaks, K im Cameron-Dominguez and a ll o f m y other fa m ily members, both big and small. I am especially grateful to two young women m athem aticians whose friendship I treasure: J illia n M cLeod and Lynnell M atthew s. J illia n , your friendship has made m y life better. T hank you for your consistent care and atte n tio n to me. Lynnell, this dissertation is not complete w ith o u t yours. Thank you fo r the many days when you provided an ear for m athem atics and th e countless tim es when you provided an ear fo r a friend.

v

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

A b s tra c t

The C atalan num ber sequence

1 ,1 ,2 ,5 ,1 4 ,4 2 ,.

n + l \ nJ

is one o f the most im p o rta n t sequences in all o f enum erative com binatorics. Richard Stanley cites at least

66

com binatorial settings where the sequence appears [7].

A m ong the numerous interpretations o f the C atalan numbers, we have the num ber o f paths restricted to the firs t quadrant o f the x 1y -plane sta rtin g a.t ( 0 , 0 ) and ending at (2 n ,0) using “ up” steps (1,1) or “ down” steps ( 1 , - 1 ) (known as D yck paths). T h is research w ill focus on a generalization o f the Catalan numbers which can be interpreted as ta king D yck paths and p e rtu rb in g the length o f the down step. One p a rtic u la r example o f this generalization is given by the sequence o f numbers known as the te rn a ry numbers,

1 ,1 ,3 ,1 2 ,5 5 ,2 7 3 ,1 4 2 8 ,7 7 5 2 ,4 3 2 6 3 ,..., — ( i n ) ,.... 2n + 1 \ n J T his sequence arises in m any natural contexts and extensions o f known results re­ lated to com binatorial objects such as paths, trees, perm utations, p a rtitio n s, Young tableaux and dissections o f convex polygons.

Hence, we use this sequence as an

im p o rta n t example o f something w hich lies on the boundary o f what is known and w hat is new. M uch o f this research is an a tte m p t to extend m any o f the known results fo r the Catalan numbers to te rn a ry and m -a ry numbers. vi

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

In this dissertation, we establish, in the setting of the tern ary numbers, several analogues to the b e tte r known C atalan num bers setting. We present an analogue to the Chung-Feller Theorem which says th a t fo r paths from (0,0) to (3n, 0) w ith step set { ( 1 , 1 ), ( 1 , —2 ) } , the num ber o f up ( 1 , 1 ) steps above the rr-axis is u n ifo rm ly d istrib u te d . We also present analogues to th e B inom ial, M o tzkin and Fine gener­ a ting functions and discuss com binatorial in terpretations o f each. In a d d ition , we establish some com putational results regarding the area bounded by tern ary paths and the num ber o f returns to the ar-axis. We also present generating fu n ctio n proofs fo r the area under D yck and te rn a ry paths, an interesting connection to weighted trees, and a conjecture fo r a closed fo rm generating function for the area under te rn a ry paths.

vn

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Table o f C ontents D is s e r ta tio n A p p r o v a l S h e e t ............................................................................

ii

D e d ic a t io n .................................................................................................................

iii

A c k n o w le d g m e n t s ................................................................................................

iv

A b s t r a c t ....................................................................................................................

vi

L is t o f F i g u r e s .......................................................................................................

x

C h a p te r 1. I n t r o d u c t i o n ...................................................................................

1

Basic D efinitions and O v e r v ie w ..................................................................

1

1.1.1

D e fin itio n s .............................................................................................

1

1.1.2

Overview

6

1 .1

1.2

1.3

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

The Catalan Numbers: Some Interpretations and Related Results

. .

8

1.2.1

C om binatorial Inte rpretation s o f the Catalan Num bers

. . . .

8

1.2.2

Statistics Related to the Catalan N u m b e rs ..................................

10

1.2.3

Functions Related to C { z ) ...............................................................

10

The Ternary numbers and Generalized Dyck P a t h s ..............................

12

1.3.1

C om binatorial Inte rpretation s o f the Ternary Num bers

. . . .

12

1.3.2

Generalized i-D y c k p a th s ...................................................................

17

v iii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1.4

The Riordan G r o u p ............................................................................................. 17

C h a p te r 2. A n a lo g u e s o f C a ta la n P r o p e r t i e s ............................................ 24 2.1

T (z ) is to N (z ) as C (z ) is to B ( z ) ................................................................... 24

2.2

The Chung-Feller Theorem for f-D yck p a t h s .............................................. 26

2.3

The M o tzkin A n a lo g u e ...................................................................................... 29

2.4

2.3.1

The Balls on the Lawn P r o b le m ..........................................................30

2.3.2

The Euler T r a n s fo r m .............................................................................. 33

The Fine A n a lo g u e ............................................................................................. 35

C h a p te r 3. R e tu r n s a n d A r e a .............................. 3.1

3.2

38

Expected Num ber o f R e t u r n s ......................................................................... 38 3.1.1

D yck P a t h s ...............................................................................................39

3.1.2

Ternary P a th s ........................................................................................... 39

Techniques fo r F in d in g Area Under Paths

................................................... 43

3.2.1

D yck P a t h s ...............................................................................................44

3.2.2

Ternary P a th s ........................................................................................... 48

C h a p te r 4. C o n c lu s io n s a n d O p e n Q u e s t io n s ............................................ 53 4.1

Sum m ary and W ork to be D o n e ......................................................................53

4.2

M ore Open Q uestions..........................................................................................54 4.2.1

Elements o f Pseudo O rder 2 .................................................................54

4.2.2

D eterm inant Sequences........................................................................... 57

4.2.3

Narayana A n a lo g u e ..................................................................................62

ix

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

List o f Figures 1.1

The five D yck paths o f length

from (0,0) to ( 6 , 0 ) .............................

2

1.2

The six Schroder paths of length 4 from (0,0) to (4 ,0 )............................

2

1.3

The preorder o f a plane tree T ......................................................................

5

1.4

P ic to ria l representation o f C (z ) =

6

1.5

The three te rn a ry paths fro m (0,0) to ( 6 , 0 ) .................................................. 13

1.6

The dot diagram o f R

........................................................................................21

1.7

The dot diagram o f C

........................................................................................2 1

1.8

G enerating fu n ctio n fo r D yck paths w ith k returns is zkC k(z ) ...................23

3.1

A plane tree, T , such th a t v {T ) = 2 2 ............................................................. 45

3.2

A plane b in a ry tree, B . such th a t to{B ) = 2 4 ...............................................46

3.3

G enerating fu n ctio n fo r vertices at height

3.4

C alculation o f A rea Under S tric t Ternary Paths and Ternary Paths

4.1

Elem ents o f Pseudo O rder 2 and T h e ir D ot D ia g ra m s ............................... 56

4.2

The determ inant sequences o f the tern ary numbers

4.3

The Narayana polynom ials, N n( x ,y )

4.4

The Narayana polynom ials in term s o f elementary f u n c t io n s .................. 63

6

1

4- z C 2(z )

1

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

is2z y C 2( z ) ............................ 47 . 51

.................................. 61

............................................................... 63

x

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

C hapter 1 Introduction 1.1 1.1.1

Basic D efinitions and Overview D efinitions

T hroughout th is dissertation, we w ill refer to objects known as paths and trees. There are m any ways to view these objects, so here we set o u t the term inology as we inten d to use it.

D e f in it io n

1 .1

. A p a th , P , o f length n fro m (xo?J/o) to (x, 3/) w ith step set S is a

sequence o f points in the plane, (x 0, yo), (# i, y i) , (ar2 , 2/2 ), • ■•, ( x n, Vn) = { x ,y ) such that a ll (x i+ i — X i,y i+ i — yf) € S. These points are called v e rtic e s . We define the h e ig h t o f a vertex, v, to be the ordinate o f that point.

We say that a path, P, is

p o s itiv e i f each o f its vertices has nonnegative height.

E x a m p le

1 .1

. D y c k p a th s are positive paths fro m (0,0) to ( 2 n, 0) with S =

{(1 ,1 ), (1, —1 )}. See Figure 1.1.

E x a m p le

1 .2

. Schroder paths are positive paths fro m (0,0 ) to (2n, 0) w ith S =

{ ( 1 , 1 ), (2 ,0 ), (1, —1 )}. See Figure 1.2. Schroder paths are counted by the big Schroder 1

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Figure

1 .1 :

The five Dyck paths o f length

6

fro m (0,0 ) to ( 6 ,0 )

/ \ Figure 1.2: The six Schroder paths o f length 4 fro m (0 ,0 ) to (4,0).

numbers, 1 ,2 ,6 ,2 2 ,9 0 ,—

D e f in it io n 1.2. A (simple) g ra p h is a triple, G = (V, E , fi), where V is a fin ite set o f vertices, E is a fin ite set o f edges and is a fu n c tio n which assigns a unique 2-element set o f vertices { u , v } , u f ^ v , to each edge e. Since 4>{e) = {u , u } is unique f o r each e, we w rite e = uv and say e jo in s u and v. A vertex u is a d ja c e n t to v i f there exists an edge e such that e = uv.

D e f in it io n 1.3. A graph G is c o n n e c te d i f f o r any two d istin ct vertices u and v, there exists a sequence vqV\ • • • vn o f vertices such that vq =

u,

vn — v and any two

consecutive term s V{ and ut-+i are adjacent. A c y c le is a sequence vqV\ ■- - vn, n > 1, o f vertices such that vq = vn, a ll other u,- ’s are distinct and any two consecutive term s are adjacent.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

D e f in it io n 1.4. A (rooted) p la n e tr e e o r tre e , T , is a connected graph w ith no cycles, where one vertex is designated as the root.

E x a m p le 1.3. We take the convention o f com puter scientists and draw o u r trees w ith the root at the top o f the tree. Pictured are the five plane trees on 4 vertices.

E x a m p le 1.4. Below is an example o f a plane tree, T , consisting o f a root labelled a, and subtrees 7 \ = { b , e , f , h } , T2 = { c } , and T 3 = {d ,g , i , j } . a

We say that b has two successors. S im ilarly, a has three successors and g has two. We say that g is the p re d e c e s s o r o f i and j . Every non-root vertex has a unique predecessor. We say that e and f are s ib lin g s . S im ila rly, i and j are siblings, as are b, c and d. A vertex with no successors is called a le a f o r e n d p o in t. Vertices h j ; h, f and c are leaves.

We define the le n g th o f a tree, T , denoted l ( T ) to be the num ber o f edges o f T. For any vertex, v, we define the d e g re e o f v, denoted deg(v), to be the num ber o f successors o f v. We define the h e ig h t o f v, denoted h g t(y ), to be the length o f th e unique “ path” fro m the root to v; th a t is, h g t(v ) = n, where uqUi • • • vn is the

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

unique sequence o f vertices such th a t v0 is the root, vn = v and any tw o consecutive term s u; and u;+i are adjacent. T h e height o f v is 0 i f and only i f v is the root. As an illu s tra tio n , we have 1{T) = 9, deg(a) = 3, and hg t(h) = 3 fo r the tree T in Exam ple 1.4. A plane m -a ry tree is a plane tree in w hich every vertex (including the ro o t) has degree

0

o r m.

E x a m p le 1.5. The five 2-ary (o r binary) trees with 6 edges.

E x a m p le

1 .6

. The three 3-ary (o r te rn a ry) trees with 6 edges.

G iven a tree, T , i t is useful to have a canonical linear ordering o f its vertices. One such ordering is called the p r e o r d e r o f T . The preorder o f T can be described (in fo rm a lly yet in tu itiv e ly ) in the fo llo w in g way. Im agine the edges o f T as wooden sticks and im agine a worm s ta rtin g at the root o f T and crawling down. I t begins towards th e leftm ost edge adjacent to the root and continues clim b ing along the edges o f th e sticks u n til it reaches th e root again (for the last tim e ).

Recording

each vertex as the w orm reaches i t fo r th e firs t tim e gives us the preorder o f T . See figure 1.3.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

5

Figure 1.3: The preorder o f a plane tree T

There is a standard bijection between D yck paths o f length 2n and plane trees o f length n. We w ill call i t the “ w orm b ije ctio n ” since i t is closely related to the preorder. Given a plane tree, T , o f length n, we can map to a unique D yck path, D . o f length 2n, by traversing T in preorder and associating an up (1,1 ) step w ith the act o f traversing an edge downward away from the root and a down ( 1 , -

1)

step w ith

traversing an edge upward tow ard the root. C all this m ap W . As an illu s tra tio n , W

(0, 0)

(10, 0)

To see th a t the map is reversible, take a D yck path, D , o f length 2n and “ collapse” D to a plane tree, T , w ith n edges in the follow ing way.

Im agine squeezing D

between your hands so th a t an up step meets its corresponding down step to form an edge o f T . Then flip T upside down.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

6 We le t D n denote the set o f a ll Dyck paths o f length 2n and A n denote the set o f all plane trees w ith n edges. I t can be shown th a t

|D n | = K | = —

We le t Cn denote the n th Catalan num ber,

( n )

( 1 .1 )

(2” ), and we call C( z) =

the Catalan generating function. One way to see ( 1 . 1 ) is the follow ing: Take the setting o f Dyck paths. A D yck path consists o f either the em pty path only or it consists o f an up step, followed by a Dyck path at height

1,

followed by a down step,

followed by a D yck path at height 0. (T his is illu stra te d in Figure 1.4.) The result o f th is observation is th a t the generating function for Dyck paths, C( z) , satisfies

C{ z) = 1 + z C \ z ) .

The quadratic fo rm u la yields C{ z ) = * ~ 2 ~

, and an application o f the Lagrange

Inversion form ula [4, 38] or T aylor series provides the n -th coefficient, ^

y

(2*)-

OR

Figure 1.4: P ic to ria l representation o f C{ z) =

1.1.2

1

+ z C 2(z)

O verview

C hapter

1

o f th is dissertation is intended to lay the foundation and p aint the

background for the rem ainder o f the document. We hope to (1) give the reader the language and no ta tio n used throughout the paper, ( 2 ) m otivate the questions asked

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

in the la te r chapters, and (3) present the tools used to answer these questions. In the previous section, we presented some basic definitions and examples and established the language used th ro ugho ut this dissertation.

In section 1.2, we discuss some

com binatorial in terpretations o f the Catalan numbers and some o f the interesting results related to those settings. In section 1.3, we describe th e sequence o f numbers known as the te rn a ry numbers. We w ill discuss the te rn a ry num bers as a base case generalization o f th e C atalan numbers and discuss the com bin atorial sim ilarities between the two. In section 1.4, we define the R iordan group and dem onstrate its u t ilit y as a com bin atorial to o l fo r solving enum eration problem s. In C hapter 2, we ask th e follow ing question: How can the results re la tin g to the Catalan numbers (i.e., th e results o f section 1.2) be extended to the generalized setting represented by the te rn a ry numbers? W h a t about m -ary numbers?

Are

there unusual com plications involved in th is extension, and, i f so, to w hat can they be a ttrib u te d ? As is the case throughout the research, we w ill use the R iordan group where appropriate to obtain and in te rp re t results. In C hapter 3, we tu rn our a tte n tio n to more q u a n tita tiv e aspects o f the combi­ n a to ria l structures studied so far. In p a rticular, we w ant to know about quantities such as th e num ber o f returns to the x-axis (for paths), the degree o f the root (for trees), and the area o f th e region bounded by a path and the x-axis. In this pursuit, as w ith others, th e ideal goal is to have a b ijective p ro o f o f th e results. I t is always interesting to know to w h at extent the R iordan group can be used to provide insight fo r the construction o f such bijections.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

8 In Chapter 4, a sum m ary o f the work is presented along w ith unanswered ques­ tions. In a dd ition , some new ideas w ith open questions are raised.

1.2

The Catalan Numbers: Some Interpretations and R elated R esults

1.2.1

Com binatorial Interpretations o f th e Catalan N um ­ bers

The Catalan num ber sequence, 1 ,1 ,2 ,5 ,1 4 ,4 2 ,...,^- (2” ) , — is probably the sec­ ond best know n sequence (a fte r the Fibonacci numbers) in a ll o f enumerative com­ binatorics. There have been numerous papers w ritte n about them . In fact, in [38], Richard Stanley lists

66

com binatorial in terpretations o f the Catalan num ber se­

quence. M ore in terpretations have been listed on his home page [36]. In Section 1 .1 ,

we saw th a t C ( z ) counts Dyck paths from (0,0 ) to (2n, 0) and plane trees w ith

n edges. In this section, we w ill introduce a few m ore interpretations and related results in order to m otivate the w ork to be done in chapters 2 and 3.

T h e o re m

1 .1

. Cn = ^ p ( 2" ) is the number of:

(i.) triangulations o f a convex ( n Jr'2)-gon into n triangles w ith n — 1 nonintersecting (except at a vertex) diagonals,

( ii.) planted (i.e., degree o f the root is 1) plane trees w ith 2 n +

2

vertices where every

nonroot vertex has degree 0 o r 2,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

9 (in .) D yck paths fro m (0,0) to (2n + 2,0) w ith no peak at height 2, (iv.) noncrossing p a rtitio n s o f [n]. (See [38])

P r o o f:

(i.) [P roof by bijection to plane b inary trees. [38]] The picture should make

clear the m ap which takes a tria n g u la tio n o f a polygon to a plane binary tree.

N ote th a t the edges o f the convex polygon P are th ic k solid lines and the vertices o f P are dots, w hile the edges o f the corresponding b in a ry tree, T , are th in solid lines and the vertices o f T are stars. Then we rota te the picture o f T and redraw it according to our convention. The root o f T corresponds to the ’’ b o tto m ” edge o f P . (ii.) [P roof by bije ctio n to plane binary trees provided in [38].] (iii.) [Proof by generating functions and by bijection provided in [26].] (iv .) [P roof by bijection to plane trees.] Each element o f [n] represents a vertex o f a plane tree w ith n + 1 vertices labelled in preorder startin g w ith the root labelled 0. Make elements o f the same block siblings and place the rem aining vertices according to the preordering o f the tree. 1 /2 /3

1/23

13/2

12/3

123

(T h is correspondence is due to Dershowitz and Zaks [6 ].)

□

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1.2.2

Statistics R elated to th e Catalan N um bers

M uch work has been done w ith respect to the enum eration o f com binatorial structures related to the Catalan numbers according to various statistics. As we w ill see in C hapter 3, the negative binom ial p ro b a b ility d is trib u tio n w ith parameters r and p, denoted n e g b in (r,p ), shows up often w ith , in m any cases, r = 2.

The

fo llo w in g theorem. [32] presents fu rth e r evidence o f this fact.

T h e o r e m 1.2. (Shapiro) (i.) The lim itin g distribu tion o f Dyck paths enumerated by number o f h ills is negbin(2, ( ii.)

The lim itin g distribu tion o f noncrossing pa rtitio n s enumerated by number o f

visible blocks is negbin( 2 , | ) .

A n oth er theorem related to the statistics o f Catalan numbers is the Chung-Feller Theorem [5, 11]. This theorem addresses the follow ing question. O f a ll paths from ( 0 , 0 ) to ( 2 n , 0 ), how m any paths have the p rope rty th a t exactly 2k (0 < k < n) o f th e ir steps lie above the x-axis?

The Chung-Feller Theorem asserts th a t the

surprising answer is c„, regardless o f k. We w ill re visit th is theorem in C hapter 2.

1.2.3

Functions R elated to C(z)

There are some generating functions w hich form interesting relationships w ith C( z) . In p a rticu la r, they are the C entral B in o m ia l function, B ( z ) 1the Fine function, F ( z ) , and the M o tzkin function, M ( z ) . In [7], Shapiro and Deutsch lis t and prove several id entities in volving the Catalan, Fine, and C entral B inom ia l functions. Here,

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

11 we describe these functions and present a few identities. D e f in it io n 1.5. The Central B in o m ia l fu n ctio n , B( z ) , is the generating fu n ctio n f o r the central binom ial coefficients, (2n) . That is,

71=0

B ( z ) counts paths from 48 [ Z, Z]

V

7

V

o (2 n ,0) w ith S = {(1 ,1 ), (1, —1 )}, often referred

to as b ila te ra l D yck paths. I d e n t it y

1 .1

D e f in it io n

. (See [7]) B ( z ) = 1 + 2z C ( z ) B ( z )

1 .6

. The Fine fu n ctio n , F ( z ) , is the generating fu n c tio n f o r the Fine

numbers, 1 ,0 ,1 ,2 ,6 ,1 8 ,5 7 ,1S6,..., (named after in fo rm a tio n theorist Terrence Fine o f C o rne ll U n iversity). rr( ~\

-

V

'

f ~n -

1 ~ V 1 ~ 4z *(3 -

The Fine numbers count D yck paths w ith no hills. A h ill is defined as an up step (sta rtin g on the a>axis) followed im m ediate ly by a down step. They also count plane trees w ith no leaf at height Lem m a

1

1.

. (See [7])

( Q fn ~ §Cn (a .) F ( z ) = D e f in it io n 1.7. The M otzkin fu n ctio n , M ( z ) , is the generating fu n c tio n f o r the M otzkin numbers 1 ,1 ,2 ,4 ,9 ,2 1 ,5 1 ,1 2 7 ,3 2 3 ,... ji *( \ V" « 1 — z — \ / l — '2z — 3z2 M ( z ) = 2 ^ m nz = --------2z 71=0

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

12 T he M o tzkin numbers have a num ber o f com binatorial interpretations. They include: 1.

the num ber o f ways o f drawing any num ber of nonintersecting chords on n points on a circle, (n = 3 pictured below)

3

2.

3

3

3

the num ber o f positive paths from ( 0 , 0 ) to (n, 0 ) using S = { ( 1 , 1 ), (1 ,0 ), ( 1 , —1)},

3. the num ber o f plane trees w ith n edges where every vertex has degree < 2.

4. the num ber o f noncrossing p a rtitio n s 7r = { B i , ..., B k } o f [n] such th a t i f B i = {b } and a < b < c. then a and c are in different blocks o f tt. I d e n t it y

1 .2

. M(z) =

Section 2.3 provides fu rth e r explanation o f the M o tzkin numbers.

1.3

The Ternary numbers and Generalized Dyck Paths

1.3.1

Com binatorial Interpretations o f th e Ternary N um ­ bers

Let T denote the set o f all positive paths fro m (0,0 ) to some (x , y) w ith S = { ( 1 , 1 ), ( 1 , —2 ) } . Let Tn,k denote the paths in T o f length n in w hich the term in al

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

13 vertex has ordinate k. so th a t T = l j n k>0 Tn,k- We w ill refer to paths in 7^,o as tern ary paths. See figure 1.5.

/\A Figure 1.5: The three tern ary paths from (0,0) to (6,0)

Let

— | T r i , f c | and f 3 n,0 —

T h e o re m 1.3. Let T ( z ) =

Then:

(i.) T(z) = 1 + zT*(z)

( iii.) t n

=

2 n + l ( n )

P r o o f: (i) Associate a z to each down step. Every tern ary path can be characterized as either the em p ty path or a path which starts w ith an up step followed by a ternary path at height

1,

another up step, a tern ary path at height

finally, a tern ary path a t height

.

Hence, T( z ) =

1

0

2

, a down step, and,

.

OR

+ z T 3(z).

(ii.) Characterizing paths ending at height k as in (i), we have th a t the generating fu nction for t n^ is in fact given by T k+1. The rest follows from the Lagrange Inversion form ula. See [4].

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

14 ( iii.) Follows fro m (ii.)

□

Em eric Deutsch has provided the follow ing e x p lic it expression for T (z ).

2

sin

T ( z ) = -------

However, in m ost cases, th e equation T ( z ) = l + z T z(z) proves m ore useful in solving problems in v o lv in g the te rn a ry numbers. The C atalan-like sequence given in Theorem 1.3(iii) is known as the ternary numbers. There are m any com binatorial objects counted by the tern ary numbers. We lis t some o f those objects here and provide a few o f the corresponding bijections. The tern ary num bers count:

( 1 ) positive paths sta rtin g at ( 0 , 0 ) and ending at ( 2 n, 0 ) using steps in

{ (1 ,1 ), (1, - 1 ) , (1, - 3 ) , (1, - 5 ) , ( 1 , - 7 ) ,

( 2 ) rooted plane trees w ith

2n

edges where

even degree (from this po in t on, referred

(3) rooted plane trees w ith 3n edges where

every node (in clu d in g the ro ot) has to as “ eventrees'” ):

every node (in clu d in g the root) has

degree zero or three (referred to here as “ tern ary trees” );

(4) dissections o f a convex 2(n + l)-g o n in to n quadrangles by draw ing n — 1 diagonals;

(5) non crossing trees on n + 1 points;

( 6 ) recursively labeled forests on [n].

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

15 For m ore d e ta il regarding these objects, see pages 92-93, 168-170 o f [38]. P r o o f:

(1) There exists a bijectio n between these paths and tern ary paths. I t is

possible to take each one o f these paths and convert i t to a unique tern ary path w ith the follow ing procedure. Given a path, P , o f ty p e (1), create a new path, ib(P). by replacing each down ( 1 , —(2s + 1)) step o f P w ith an up (1,1) step im m ediately followed by s + 1 down (1,-2) steps. For example,

ib

(4,0)

( 6 ,0)

(0,0)

Then ib(P ) is a positive path from (0,0 ) to (3 n,0) consisting o f steps in { ( 1 , 1 ), ( 1 , —2 )}. I t is easy to see th a t 0 is invertible. For example, ib - 1

(6 ,0)

( 0 ,0 )

(2)

(0,0)

(4,0)

(due to Deutsch, Feretic, and Noy [8 ]) There exists a b ijectio n between even

trees and te rn a ry trees. Given a nonem pty te rn a ry tree T ,

T

=

recursively define an even tree T ' by

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

16 (3) There exists a bijection from tern ary paths to tern ary trees. F irst, take a tern ary tree and label the children o f every vertex o f degree 3, U, U, D, from le ft to rig h t. Then read the labels o f the tree in preorder. Let U correspond to (1,1) and D correspond to ( 1 ,- 2 ) . (4) P ro o f by bijectio n to ternary trees. (See proof o f Theorem l. l ( i ) . )

(5)

P ro o f by bije ctio n to ternary trees. A noncrossing tree is defined as a tress on

a lin e a rly ordered vertex set where whenever i < j < k < I, then i k and j l are not both edges. Equivalently, a noncrossing tree is a tree whose vertices are n points on a circle labelled { 1 , 2 , . . . , n } counterclockwise and whose edges are rectilinear and do not cross. The p ictu re below represents the bijectio n between noncrossing trees and ternary trees. N ote th a t i, 1 < i < n, is the smallest labelled vertex adjacent to the vertex labelled

1.

( 6 ) See [38], exercise 5.45, p. 92, and solution 5.45, p. 137.

□

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

17

1.3.2

Generalized t- Dyck paths

A g e n e ra liz e d f - D y c k p a th is a positive p ath fro m (0,0 ) to ((£ + l) n , 0) w ith S =

{ ( 1 ,1 ), (1, —f ) } .

A generalized 2-D yck path is a tern ary path, w hile a

1-

D yck pa th is a D yck path. To a large extent, th is research focuses on the follow ing question: O f a ll the known properties o f D yck paths (plane trees) and the techniques fo r proving them , which ones can be ’’easily” extended to generalized f-D yck paths? Furtherm ore, which ones can not and why?

1.4

The Riordan Group

The R iordan group is used extensively throughout th is research as a com bina­ to ria l to o l fo r solving enumeration problem s. As the recent work o f Sprugnoli et al. demonstrates, there are also m any interesting questions to be answered about the R iordan group itself, a few which are explored here.

D e f in it io n 1.8. A n infinite lower triangu lar matrix, L = ( ln,k) n k > 0 m a t r ix i f there exist generating functions g(z) = ^ 2 g nzn, f i z) = f i = £ 0 such that

/ n>0

=

gn and

X )n>fc U,kZn =

a R io r d a n fnZn ■ fo =

0

,

g(z) { f ( z ))k-

From the definition it is clear th a t a R iordan m a trix L is com pletely defined by the functions g(z) and f ( z ) , hence L is called R io r d a n and we w rite L =

{ 9 ( z ) , f { z ) ) i or sim p ly L = ( g , f ) .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

18 E x a m p le 1.7. P = (^ 3 :, y f r ) , also known as Pascal’s Triangle 1 1 1

P =

E x a m p le

1 .8

1 2 1

1 3 3 1 1 4 6 4 1 5 10 10 1 6 15 20 1 7 21 35

1 5 1 15 6 1 35 21 7 1

. Express the Riordan m a trix ( C ( z ) , z C ( z ) ) in terms o f paths. 1

C = (C (z ),z C (z )) =

1

1

2

2

1

5 14 42 132

5 14 42 132

429

429

3 9 28 90 297

1

4 14 48 165

1

5

1

20

6

1

75

27

7

The firs t colum n o f C represents, o f course, the num ber o f D yck paths fro m (0,0) to (2 n ,0 ). The second colum n represents positive paths from (0 ,0 ) to (2n — 1,1) using steps in { ( 1 , 1 ), ( 1 , —1 )}.

In general, the n , k - th entry o f C represents the

num ber o f positive paths from ( 0 , 0 ) to ( 2 n — k, k) using steps in { ( 1 , 1 ), ( 1 , —1 )}. We prove th is by observing th a t since C is Riordan, the fc-th colum n o f C has generating fu n ctio n zkC k+1(z).

B u t th is is precisely the generating fu n ctio n for

D yck paths ending at height k , (i.e., having te rm in a l p o in t ( 2 n — k : k). T he next theorem gives us an expression in term s o f generating functions fo r the (weighted) row sums o f R iordan m atrices.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

19

1

E n>oa^ n ' Then

J

T h e o re m 1.4. (G etu et al. [13]) Let L = (g , f ) be a Riordan m atrix and ^ ( 2 )

CEO Cl

6

l

62

«3

O* CO

L ■ U4

b4

where B ( z ) = ^ n>Qbnzn satisfies B ( z ) = g ( z ) A ( f ( z ) ) . (■ denotes m a trix m ultipli­ cation.)

N otice we switch freely between colum n vectors and th e ir generating functions. E x a m p le 1.9. Apply Theorem 1.4 with L = C fro m example 1.8 and A (z ) = 1

C

1 - 2

1

1

2

2

1

5 14 42 132 429

5 14 42 132 429

3 9 28 90 297

1

4 14 48 165

( 1.2)

1

5

1

20

6

1

75

27

7

1

2 5 14 42 132 429 C(2) 1

~ 2(7(2)

C 2( 2)

(1.3)

(1.4) (1.5)

Equality (1.5) follows fro m the fa ct that C (z ) = 1 + z C 2(z) implies C (z) =

l-zC(z) ■

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

20

Given two R iordan m atrices, L \ = (gi(z),

and L 2 =

(9 2 ( 2 ), f 2 (z)), it

turns out th a t the product o f L \ and L 2 (under usual m a trix m u ltip lic a tio n ) is also Riordan. In fact, L 1L 2 = (g ,iz )g 2{ h { z ) \ f 2 { h { z ) ) ) Furtherm ore, (i) m a trix m u ltip lic a tio n is associative, (ii) the id e n tity m a trix I = ( 1 , 2 ) is Riordan, and (iii) every Riordan m a trix L — (g ( z ) , f ( z )) has a Riordan inverse, L ~ l — ( ^ /(J jT f ( z))i where f ( z ) is the com positional inverse o f f { z ) . Hence, the set o f Riordan m atrices, 7£, form s a group under m a trix m u ltip lic a tio n . Rogers [29] showed th a t every element,

2

n+i.A:+i, such th a t n > 0, o f a Riordan

m a trix , R, can be expressed as

• ^ n + l, fc + l

Xn,k T

® l * ^ 7 l . f c + l ” t“ ® 2 2 '7 1 ,fc + 2 “ I” ® 3 * ^ 7 l, f c + 3 “ h

• • •

where {a j}? L 0 is a sequence th a t is independent o f n and k. The sequence {o j}j2 .0 ’s referred to as the ,4-se quen ce o f R and its generating function is denoted Ar ( z) . S im ilarly, we have th a t the elements o f the first colum n o f R (excluding £ 0 ,0 ), can be expressed as

£ 71+ 1,0 =

where

2 o £ ti,0 +

~ l £ ? i, l +

Z2X n ,2 +

ZzXnS +

. . .

is a sequence th a t is independent o f n and k. The sequence { 2 j } ° l 0 is

referred to as the Z -s e q u e n c e o f R and its generating fu nction is denoted Zr (z) . We can view th e A - and Z-sequences o f a Riordan m a trix , R, p ic to ria lly as what we refer to as the d o t d ia g r a m o f R. See Figure 1.6.

E x a m p le 1 .10. The dot diagram, o f C = ( C ( z ) , z C ( z ) ) is given in Figure 1.7.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

21

20

21

22

—

ao

O

O

O

...

,

ai

o

a2

o

X

o

a3

o

X

Figure 1.6: The dot diagram o f R i

i

i

...

o

o

o

•••

i ,

0

i

i

o

o


= ]

= E

(z T 3 ( z ) ) ^

1 2

1 3 1 4 1 5 1 6 1 7

LfJ

(3-1)

/

1 3 6 10 15

k ~ j

( * - r 3(-.))t - 3 ( - . r 3(*-))‘ - 4 (*-t 3M ) ‘ - s 1 4 10

(3-2)

( z r 3M ) ‘ - 7 1

(z T 3 (z))

i

(3.3)

3=0

H aving the necessary generating functions for vertices at height k, we are now able to calculate area for te rn a ry and s tric t tern ary paths. See Figure 3.4. C urrently, we do not have a “nice” closed form fo r the generating functions above. We would like to use these generating functions to fo rm a manageable R iordan or

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

51

1 0 1 0 3 0 12 0 55 0 273

1 2 7 30 143 728

0 1

0 3

2 1 4 18

88 455

2 13 75 427

1 9 61 383

3 31 23S

1 15 140

3 4 1 58 22

21 144 981 6663

5 5

1

6 7 8

0 1 1 5 25 130 700

1 6 33 182 1020

2 2 17 115 728

1 11 86 598

3 37 321

1 17 177

3 4 4 1 66 24

5

6 7

0 3 27 207 1506 10692

8

Figure 3.4: C alculation o f Area Under S trict Ternary Paths and Ternary Paths

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

52 R iordan-like m a trix .

Given such a m a trix , we could m u ltip ly it by the Riordan

vector ( yzt ) i the product being the vectors (a®)£L0 and (c £ )£ i0. We saw in Section 3.2.1 th a t there is a correspondence between th e weight of bin a ry trees and area under D yck paths. I t is n a tu ra l to ask i f there is a sim ilar correspondence for te rn a ry trees and tern ary paths. I t tu rn s out th a t indeed such a correspondence exists. Let W n denote the set o f a ll te rn a ry trees w ith 3n edges.

Conjecture 3.1. F o r a ll n € N,

] T u,(W) = J 2 < T ) Wt=Wn TeTn I f we le t t ( k , n ) denote the to ta l num ber o f vertices at height k in W „ and r ( y , z)

= ' ^ 2 l { k , n ) y hzn, then we can easily show (see p ro o f o f Theorem 3.3) th a t L u i d y( (y ,~ ))U i

-

ZzTZ^ ( i _ 3 z T 2(z ))2

=

3 2 + 27 z2 + 207s3 + 1506z4 + 10692z5 + . . .

which agrees w ith the area function for tern ary paths up to the first several terms. Hence, it remains to prove th a t

the area function.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

C hapter 4 C onclusions and O pen Q uestions

4.1

Sum m ary and Work to be D one In C hapter 2, we established, in the setting o f the te rn a ry numbers, several

analogues to th e be tte r known Catalan numbers setting.

We presented an ana­

logue to the Chung-Feller Theorem [5, 11] which says th a t fo r paths fro m (0,0) to (3n, 0) w ith step set {(1 ,1 ), (1, —2 )}, the num ber o f up (1 ,1 ) steps above the ar-axis is u n ifo rm ly d istrib u te d . We also presented analogues to the B inom ial, M o tzkin and Fine generating functions and discussed com binatorial in terpretations o f each. F u rth e r study o f the functions should lead to m ore interpretations. In p a rticu la r, fu rth e r study in to the existence o f an Euler Transform analogue and analogues to the F in e /C a ta la n /B in o m ia l id entities [7] is needed. In C hapter 3, we established some com putational results regarding area and the num ber o f returns in the context o f ternary paths. We concluded th a t the num -

53

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

54 ber o f returns for generalized t-D yck paths approaches a negbin(2,

d istrib u tio n .

F urth e r study w ill address rem aining questions, such as the com binatorial signifi­ cance o f th e param eter 2. We also presented generating fu n ctio n proofs for the area under D yck and tern ary paths, a connection to weighted trees, and a conjecture for a closed fo rm generating function for the area under te rn a ry paths. Future work regarding area under paths w ill include fin d in g generating functions for the area under generalized t-D yck paths and provid ing b ije ctive proofs.

4.2

M ore O pen Questions

The rem ainder o f the dissertation addresses some open research questions which, fo r the sake o f order, have been grouped apart fro m th e preceding results.

The

reason fo r th e separation is th a t the m ate rial th a t follows, although related to the ideas covered so far, has connections to other topics in com binatorics not necessarily emphasized in the previous chapters.

4.2.1

E lem ents o f Pseudo Order 2

A n element R o f th e Riordan group is said to have pseudo order two, or order 2 *, i f R M has order two, where M = (1, —z). A prim e example o f an element o f order 2* is the generalized Pascal’s triangle m a trix , survey shows th a t i t is the case for some b th a t instance,

$2

= ( 1_12-^ , p~2f)- ). A n in itia l = B M B ~ l M fo r some B . For

= C q M C ^ M , where Co = ( C 2 ( z ) ,z C 2 (z)).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

I

We would like to fin d com binatorial interpretations o f such elements o f order 2*. We know fo r example th a t elements o f the form

have order 2*. One

m ig h t ask, i f R has ' der 2*, is it the case th a t R =

fo r some B? I f so,

w hat is the com binatorial relationship between R and B? I t turns out th a t we can a ffirm a tive ly answer the first o f these questions fo r a special class o f matrices, and, in some cases, characterize B given a certain R.

T h e o r e m 4 .1 . Let '1 + A C

0b,X,c,S

—

i /

( ^

)

-

cz /*•> ( Xz2 \ l-bz^ \{l-bz)2)

( f ! ^ ) Sz2

!

I Xz2 ^ \(l-bz)2J

1 — 2bz 1 — 2bz

Then

(i-) ©6,>,e,5 has order

2

*.

( ii.) There exists a R iordan m a trix B such that Qb,x,t,s = B M B _ 1 M .

C o r o lla r y 4 .1 . #6 = B M B ~ l M where f B =

2n

1 — bz — \ / l — 2 bz + (b2 — An)z

2n — k — (2 n — k)bz + k y / l — 2bz + (b2 — An)z 2

2 nz

f o r some n and k.

C o r o lla r y 4 .2 . Qi x , 0 = j

(

y ( 2 X + t b ) z - c + e y / l - 2 b z + ( b 2 - 4 X ) z 2 1 -2 6 2 ’ ^ b z

J

where n , B =

1 - (2c + b)z + y / l -

I — bz — y / l — 2bz + (b2 — 4 n )z 2 2

bz + ( b * - 4 \ y

2

nz

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

56 D ot diagram o f A *

D ot diagram o f A

Note: S (z ) = 1 3~ bers, 1,3,11,45,...

~~~ is the generating fu n ctio n fo r the litt le Schroder num ­

Figure 4.1: Elements o f Pseudo O rder 2 and T h e ir D ot Diagrams

However, there exist elements o f order 2* w hich are not o f the class 06,a,c,5- A n example is the B e ll subgroup element (1 + z M ( z ) ,z ( 1 + z M (z ) )) , where M (z ) is the M o tz k in generating function. T his m a trix counts directed anim als and the rows approach negbin{2, | ) . O ther examples are N kw anta’s R N A triangle n

f 1—

R== V

Z + Z2

— y / l — z — 3z 2 _ 1 — z + z 2 — \ / l — z — 3z2^

2*

,Z

2^

)

w hich counts secondary R N A structures [32] and C = (2C (z) — 1, z(2 C (z) — 1)) which counts D yck paths w ith h ills th a t are one o f tw o colors. In the case o f the la tte r, we have th a t C = B M B ~ XM where B = (C (z ),z C (z )). (See Section 1.4) As fo r R, i t is s till unknown whether there exists B such th a t R = B M B ~ l M .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

4.2.2

D eterm inant Sequences

Yet another remarkable p rope rty o f the Catalan numbers has to do w ith its de­ te rm in a n t sequences. G iven a sequence {