Combinatorial Interpretations of Hankel Matrices and Further Combinatorial Uses of Riordan Group Methods

Table of contents :
DISSERTATION APPROVAL SHEET ......................................................... ii
DEDICATION ........................................................................................................ iii
ACKNOWLEDGMENTS .................................................................................. iv
ABSTRACT ............................................................................................................ vi
LIST OF FIGURES .............................................................................................. x
LIST OF TABLES ................................................................................................ xii
CHAPTER
1. Introduction ..................................................................................................... 1
1.1 Hankel Matrices and Determ inants ......................................................... 1
1.2 The Riordan Group ...................................................................................... 4
1.3 Directed Graphs ................................................................................................. 10
2. Schroder Numbers and Hankel Determinants ..................................... 12
2 .1 Schroder Numbers ........................................................................................ 12
2.2 Hankel Determinants of Schroder Numbers .............................................. 15
2.3 Combinatorial Interpretations ........................................................................ 29
3. Generalization of Schroder Numbers R esu lts ....................................... 31
3.1 m -adm issible ..................................................................................................... 31
3.2 4-admissible Sequence ..................................................................................... 33
3.2.1 4,-admissible Translation ...................................................................... 35
3.2.2 Schroder Analog ................................................................................... 36
3.2.3 Free Bishop M oves ............................................................................... 37
3.2.4 Up Bishop Moves ................................................................................. 40
3.3 Bijective proofs ................................................................................................. 43
3.3.1 4-admissible Translation Schroder Analog .......................... 43
3.3.2 Free Bishop Moves • Schroder A nalog ....................................... 47
3.3.3 Up Bishop Moves Schroder Analog ....................................... 51
3.4 Main R esult ....................................................................................................... 58
3.5 Additional Combinatorial Interpretations ..................................................... 59
4. Disjoint Motzkin Path Systems .................................................................. 62
4.1 Disjoint paths and Hankel Determinants ..................................................... 62
4.2 Motzkin Paths ................................................................................................... 63
5. Conclusion ........................................................................................................... 72
5.1 Open Questions
REFERENCES .........

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HOWARD UNIVERSITY C om binatorial Interpretations o f H ankel M atrices and Further Com binatorial U ses of R iordan G roup M ethods

A Dissertation Submitted to the Faculty of the Graduate School

of

HOW ARD U N IV E R SIT Y

in partial fulfillment of the requirements for the degree

D O C TO R OF PH IL O SO PH Y

Department of Mathematics

by

Lynnell Sherri M atthew s

Washington, D.C. December 2001

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UMI Number: 3066512

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H O W A R D U N IV E R S IT Y G R A D U A T E SCHOOL D E P A R T M E N T OF M ATH EM ATICS DISSERTATION COMMITTEE

Neil Hindman. Ph.D. Chairperson

g ' . , . _ Paul Peart, Ph.D

XcUiA V . Louis W. Shapiro, Ph.D.

Wen-Jin Woan, Ph.D.

Darla Krefter, Ph.D. Assistant Professor of Mathematics Gettysburg College

Louis W. Shapiro, Ph.D. Dissertation Advisor Candidate: Lynnell Sherri Matthews Date of Defense: November 29, 2001

ii

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D edication

To my mother, the late B ernice E velyn Johnson, who continues to a have posi­ tive impact on my life.

To my son, Howard Le R oy M atthew s, III, may this inspire you to achieve the seemingly unattainable.

111

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A cknow ledgm ents In the words of the old African proverb, “It takes a village to raise a child...,” I know that I have truly been blessed with a nurturing village. First and foremost, I thank God for surrounding me with the people who played an integral part in this achievement. I thank my husband, Steven Andre Matthews, for his undying love, encourage­ ment, support and tolerance. I thank my father, Richard A. Johnson, and my sister, Bernita A. Johnson, for their financial and emotional support. I thank the Johnson, Washington and Matthews families and my dearest friends Monica Holbeck, Tina Laury and Robyn Magruder-Matthews. A special thanks to Jacqueline Matthews who is solely responsible for me being able to finish my graduate program without worrying about my son in her care. In the mathematics community, my primary supporter has been my advisor, Louis W. Shapiro. I am grateful for so many things but primarily for his belief that I was capable of achieving this ominous goal. I also thank him for teaching me that the solution to a problem does not necessarily end with a proof, th at is, a proof is just the beginning of the solution for a more general problem. I am indebted to the mathematics department at Howard University under the leadership of Dr. Joshua Leslie, Chairman. I have forged my way through a sea of knowledge being guided along by an exceptional group of mathematics professors. I thank Clemente Lutterodt, Francois Ramaroson, Walter Miller and Thiery Robart

iv

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for their encouragement and support. I also thank Cora Sadosky and Louise Raphael who are examples that woman can do mathematics well. A special thanks for the support of the Howard University Combinatorics Seminar Group and my committee members for their insightful input in preparing this disseration: Louis Shapiro, Seyoum Getu, Paul Peart, Wen-Jin Woan, Neil Hindman and Darla Kremer.

I

also thank my peers in the graduate program, especially Barbara Tankersley and Basirrou Diatta. I have been surrounded by extraordinary educators throughout my academic career. I thank Lydia Bowen, my 8th grade Geometry teacher, who saw my poten­ tial and inspired me to major in mathematics. I thank M artha Siegel my professor and mentor at Towson University. I thank the mathematics department at Towson University for building a foundation upon which I have drawn upon throughout my graduate study. I thank Mary Johnson who was instrumental in my decision to pursue graduate study. I have also been fortunate to have worked with dynamic mathematics departments at Howard Community College, Bowie State University and Gettysburg College. I thank each of my present and former collegues for their contributions to my professional growth. Last but not least, I thank my mathematical sibling, Naiomi T. Cameron. We have been adjoined at the hip by Combinatorics and we have become better m ath­ ematicians as a result of it. I can not thank you enough for the countless hours of sharing which was not always about mathematics but was always invaluable.

v

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A bstract

Random walk enumeration is a key concept in enumerative combinatorics. Several well known sequences have been used to enumerate various random walks such as the Catalan numbers, Motzkin numbers and Schroder numbers. These enu­ merations have applications in queuing theory, sorting and searching algorithms. Hankel matrices have applications in complex analysis and approximation theory. This research will address the Hankel matrices of the well known combinatorial sequences related to the Schroder numbers. Motivated by the Schroder numbers results, the research solves other related enumeration problems. For the sequences that arise, the research provides additional combinatorial settings which yield these numbers. In addition to the enumeration problems, the research explores the rela­ tionship between Hankel matrices and disjoint path systems.

vi

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Table o f Contents

D ISSER TA TIO N APPRO VAL S H E E T .........................................................

ii

D E D IC A T IO N ........................................................................................................

iii

A C K N O W L ED G M EN TS ..................................................................................

iv

A B ST R A C T ............................................................................................................

vi

LIST OF FIG U R E S ..............................................................................................

x

LIST OF TA BLES ................................................................................................

xii

CHAPTER 1.

2.

Introduction .....................................................................................................

1

1.1

Hankel Matrices and D e te rm in a n ts .........................................................

1

1.2

The Riordan G ro u p ......................................................................................

4

1.3

Directed G raphs.................................................................................................10

Schroder N um bers and Hankel D eterm in an ts .....................................

vii

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12

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

12

2.2 Hankel Determinants of Schroder N u m b e rs..............................................

15

2.1

Schroder Numbers

2.3 Combinatorial In te rp re ta tio n s........................................................................ 29

3. G eneralization o f Schroder N um bers R e s u l t s .......................................

31

3.1 m -a d m is s ib le .....................................................................................................31 3.2 4-admissible S eq u e n c e .....................................................................................33 3.2.1 4,-admissible T ran slatio n ......................................................................35 3.2.2 Schroder Analog

...................................................................................36

3.2.3 Free Bishop M o v e s ............................................................................... 37 3.2.4 3.3

Up Bishop M oves................................................................................. 40

Bijective p r o o f s .................................................................................................43 3.3.1

4-admissible Translation

3.3.2

Free Bishop Moves• Schroder A n a lo g .......................................47

3.3.3

Up Bishop Moves

Schroder A n a lo g .......................... 43

Schroder A n a lo g ....................................... 51

3.4

Main R e s u l t .......................................................................................................58

3.5

Additional Combinatorial Interp retatio n s.....................................................59

4. D isjoint M otzk in P ath System s ..................................................................

62

4.1

Disjoint paths and Hankel D e te rm in a n ts ..................................................... 62

4.2

Motzkin P a th s ................................................................................................... 63

5. C onclusion ...........................................................................................................

viii

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72

5.1

Open Questions

R E FE R E N C E S .........

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List o f Figures 1.1

Example of LDU-decom position................................................................

3

1.2

General Riordan Group Element, (g( z ) , f ( z ) ) ..........................................

5

1.3

Example of Riordan M atrix (C(z2), zC( z2))

5

l.-l

Picture Representation of the Catalan Triangle.......................................

6

1.5

Dot Diagram for Catalan Triangle, (C(z2), z C( z 2))

9

1.6

Example of Disjoint P ath Systems Using Motzkin Paths

2.1

Small Schroder Numbers Lattice Path E n u m eratio n .................................13

2.2

Small Schroder Numbers State Diagram E num eration ..............................14

2.3

Zebras W ith Semiperimeter n = 3 .................................................................14

2.4

LDU-decomposition of D \ .............................................................................27

2.5

Decomposition of D\ = A D fAT ................................................................... 28

2.6

Example of 2-admissible matrix, Ls

2.7

Illustration of second row of L s ................................................................... 29

2.8

Illustration of row multiplication of rq * r 2 in L s .......................................30

3.1

Dot Diagram for matrix yielded from m-admissible sequence................ 33

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

........................11

.......................................................... 29

x

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xi 3.2

\-adm issible Translation P a t h s ................................................................35

3.3

Schroder Analog Picture Representation of g ( z ) ....................................36

3.4

4-admissible Schroder Analog Lattice Path E n u m eratio n ................... 37

3.5

Free Bishop Moves Picture Representation of g ( z ) ................................ 38

3.6

Free Bishop Moves Lattice P ath E n u m e ra tio n .......................................39

3.7

Up Bishop Moves Lattice P ath E num eration ..........................................40

3.8

Down Bishop Moves Lattice Path E n u m e ra tio n ....................................41

3.9

Dot Diagram for Down Bishop Moves M atrix..........................................42

3.10 Schematic for Bijective P r o o f s ................................................................... 43 3.11 Illustration of 4-admissible Translation* to Schroder A n a lo g ............. 45 3.12 Illustration of Schroder Analog to 4-admissible T ra n sla tio n * ............. 48 3.13 Illustration of Free Bishop Moves to Schroder A n a lo g ..........................49 3.14 Illustration of Schroder Analog to Free Bishop M o v e s ..........................51 3.15 Illustration of Up Bishop Moves to Schroder A n alo g ............................. 53 3.16 Illustration of Schroder Analog to Up Bishop M oves.............................54 3.17 m-admissible State Diagram E n u m eratio n .............................................60 4.1

Illustration of “Interior” of Disjoint Motzkin Path S y s te m s ................ 65

4.2 t = 0 Disjoint Motzkin P ath S y s te m s ....................................................... 65 4.3 t =

1

Disjoint Motzkin P ath S y s te m s ....................................................... 6 6

4.4 t = 2 Disjoint Motzkin P ath S y s te m s ....................................................... 67 4.5 t = 3 Disjoint Motzkin P ath S y s te m s ....................................................... 69

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List of Tables 1.1 Summary of Aigner Results [1 |.....................................................................

4

2.1 Summary of 6 ^ LDU-decomposition R e s u l t s ...........................................

16

2.2 Summary of S* LDU-decomposition R e s u l t s ...........................................

16

2.3 Construction of Sequence from Hankel determ inants. |/ / “| and |//*|

. 22

3.1 m-admissible S eq u en ces.................................................................................. 34 3.2 Bijective Correspondence for \z6\g(z) = 2 9 .................................................. 55 4.1 Motzkin Path R esults........................................................................................ 64

xii

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Chapter 1 Introduction This chapter gives a brief description of the basic tools needed to establish the foundation upon which this dissertation is developed. We begin with the definition of Hankel matrices. We then review the Riordan group techniques which will be used throughout the dissertation. We also give a brief introduction to directed graphs as the ground work for Chapter 4.

1.1

Hankel M atrices and D eterm inants

Hankel matrices, named for Herman Hankel (1839-1873), have been explored from a combinatorial perspective by such mathematicians as Aigner [1 ], Getu, et al. [9], Ehrenborg [6 ], Peart, et al. [17], [19] and Radoux [21] - [23]. The definition of a Hankel matrix is given below: Definition 1.1. Given a sequence

the Hankel m atrix, H kn is the n by n

matrix whose (i , j ) th entry is al+j +^, where the indices range from 0 to n — 1. Thus we have,

1

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H° =

\

Oq

d i

02

O3

a 1 a2 02

a 2 a3 a 4 as 02 a 4 as 0 6

0,1

O2

03

Cl4

0 ,2

03

d .\

C I5

03

a\ as oq

Hi =

a4

a 2 n -2 )

05

06

a4

07

V

02n-l

/

Hankel matrices, also referred to as striped matrices in [3], arise in applications, including stochastic processes, time series analysis and digital filtering. They can also be used to localize poles of analytic functions in Complex Analysis [1 1 |. A natural question th at arises when working with matrices is, "What is the determinant?” A fundamental property of calculating determinants is the product property of determinants, that is, det(AB) - (det /i)(det B). Decomposing a matrix into the product of other matrices often simplifies the calculation of the determinant. One way of decomposing a matrix is LDU-decomposition described in the following definition: D efinition 1.2. The L D U -decom position of a matrix, A, is A = LDU where L is a lower triangular matrix with main diagonal entries equal to 1, U is an upper triangular matrix with main diagonal entries equal to 1 and D is a diagonal matrix with main diagonal elements nonzero, [f A is symmetric, then U = LT. In order to decompose a matrix in this way it is sufficient th at the matrix be positive definite. If we have Hankel matrices which are positive definite, we are

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3 able to find their LDU-decomposition and thus calculate the determinant via the product property. Since H* is symmetric, we also have U = LT. Figure

1.1

is an

example of the LDU-decomposition of a 3 x 3 Hankel matrix.

/I [1 \ 3

13

\ / 1 0 0 \ / 1 0 0 \ / 1 1 3 \ 3 11 1 = 1 1 1 0 I I 0 2 0 1 I 0 1 4 I 11 45 / \ 3 4 1/ \ 0 0 4 / \ 0 0 1/ Figure 1.1: Example of LDU-decomposition

Several combinatorial sequences have been used to form Hankel matrices and their respective determinants have been calculated. Shapiro noted in his paper, “A Catalan Triangle” [25] that the Hankel matrices, / / “ and

formed from the

sequence of Catalan numbers have determinants equal to 1 . Expanding upon that idea in ‘“Catalan-like Numbers and Determinants” [1 ], Aigner computed the Hankel determinants for a family of sequences whose detH% = 1 while the detH \ satisfied a recurrence relation. Table 1.1 is a summary of some of the results presented by Aigner. Radoux reiterated the Catalan number result in his recent paper, “Addition formulas for polynomials built on classical combinatorial sequences” [23]. In two of his earlier papers [2 1 ] and [22], Radoux computed the Hankel determinants formed from derangement numbers and Bell numbers. Aigner [2] and Ehrenborg [6 ] both provide alternative proofs of the Hankel determinants of matrices formed using the Bell numbers. Peart [17] calculated Hankel determinants, det / / “, for various

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4

Table 1.1: Summary of Aigner Results [1 ] Sequence Catalan Numbers 1,1,2,5,14,... Motzkin Numbers 1,1,2,4,9,21,... Hexagonal Numbers 1,3,10,36,137,543,... Binomial Numbers 1,3,10,35,126,... (2nn+1)

detH®

de t Hx

1

1

1

1

1

1 ,0

,- 1 ,- 1 ,0 , 1 mod

6

3,8,21,55... F> 2n+2 (the Fibonacci number) 3,5,7,... 2n + 1

sequences including the Schroder numbers using Stieltjes matrices. In [15], Mays and Wojociechowski look at the determ inants of Hankel matrices of Catalan numbers via a counting result for disjoint path systems, a result which goes back to GesselViennot [7] and Lindstrom [14].

1.2

The Riordan Group

The Riordan group, named for John Riordan in [8 ], is a subgroup of the group of lower triangular matrices with m atrix multiplication as the group operation [8 ]. Riordan group elements have combinatorial significance because of their connection to generating functions, a basic tool in enumerative combinatorics. To form a Ri­ ordan matrix you need two generating functions namely, g(z) and f ( z ) , where g(z) starts with 1 and f ( z ) starts with a nonzero term. The first column corresponds to g(z) and each subsequent column is formed by multiplying the previous column by f ( z) . So each column corresponds to a generating function of the form g ( z ) f k(z) as

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5 depicted in Figure 1.2.

(

\ 9 gf

g f 2 •••

\ :

/

Figure 1.2: General Riordan Group Element, ( g(z). f (z))

The Riordan group elements are of particular interest because each of the columns can provide information relating to a particular combinatorial setting. In other words such a matrix gives you a more complete picture. For example, the Catalan triangle in Figure 1.3, is a Riordan matrix formed using the Catalan generating function, C( z 2) =

0fln^n,

g0 = 1 and f ( z ) = £]„>i f nz n, / , ^ 0, and we denote M = (g(z), f(z)). Thus the Catalan triangle in Figure 1.3 is denoted (C(z2). z C( z 2)). A Rior­ dan matrix th at can be interpreted combinatorially can also be manipulated within

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the group structure to provide further combinatorial applications. Riordan group methods also have uses in proving and inverting combinatorial identities as shown by Getu et al. in [8 ]. The Riordan matrices form a group under matrix multiplication w ith the follow­ ing properties: P roperty 1.3.1 (M atrix M u ltiplication). (g(z), f (z)) . (h(z), l(t)) = (g(z)h(!(z)), ((/(;))

For example,

( = where 4z 3 -I- 9z4 +

1

- z - vT -

2z

- 3z 2 1 -

2

- VI 2z

2

z - 3z2

(A /(z),z(M (z))) ^-j-) is the Pascal m atrix and M(z) = 2 1 z5...

1

*

= ]_ +

2

2z2 -f

is the generating function for the Motzkin numbers. We can

interpret this multiplication as adding in level steps of length one at any step on paths in C(z2). Therefore, the resulting Motzkin triangle represents the number of lattice paths from (0 , 0 ) to (n, k) using the steps {( 1 , 1 ), ( 1 , —1 ), ( 1 , 0 )} up, down and level, respectively.

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P roperty 1.3.2 (M ultiplication b y column v ector). If(a0. ci\, a2. .. ,)T is a col­ umn vector with generating function A(z) then

(g(z)J(z))*A(z)=g(z)A(f(z)) For example, to find the row sums of (C(z2) , z C ( z 2)) we multiply by A(z) = ■j— (1 ,1 ,1 ... .)r . That is, (C(!*),zCtf))*A(z)

=

C(22)/1(2C(z2))

-

(rrEpr) 1 - 2 2 - y/l - 4Z* 4z 2 - 2z v/1 - 4z 2 + 7 7 ( V I - 4z2 ~ 0

o

1,1,2,3,6,10,20,..., Q ^ , . . . , n > 0 .

Property 1.3.3 (Inverse).

\ 9{f {x))

,7(x)), )

where f is the compositional inverse of f . For example, ( C( z2), zC( z2))~l = ( 7 7 7 7 , 7 7 7 7 ). P roperty 1.3.4 (Id en tity E lem en t). The group identity element is

/ = (!,*). Another way to characterize Riordan group matrices is by dot diagrams which is a description of a recursive relationship between the entries of the matrix. A dot

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9 diagram can be used to form the entries of the matrix. It can also be used to deter­ mine the generating functions g(z) and f(z). The Riordan matrix (C(z2), z C( z 2)) in Figure 1.3 has the following dot diagram:

Figure 1.5: Dot Diagram for Catalan Triangle, (C(z2), z C( z 2))

To illustrate using dot diagrams to find the generating functions for g(z) and f{z), we use the Riordan group definition. The dot diagram for g(z) g depicts aj,o =

which corresponds to the equation: g = I + zgf. The dot diagram

for f ( z ) • / depicts aij = a.*_x,j_i + a j_ ij+l, i >

1

which corresponds to g f k =

z { y f k~ l T 9 f k+l)- We can use these equations to derive the closed forms for g(z) and f(z).

1 and s 0 = 3 and s* = 3. k > 1, then the following Lemma shows that these values correspond to the lower triangular matrices for and S^, respectively. L em m a 2.1. The lower triangular matrix, L, from the LDU-decomposition of ei­ ther of the Hankel matrices

or

formed from the small Schroder numbers is

2-admissible. Proof: If a m atrix satisfies recursion (2.1) then it is 2-admissible. Let s 0 = 1 and Sfc = 3, k >

1

in (2 . 1 ). Using the Riordan group dot diagram technique, we have the

following

g =

1

+ zg + 2 z g f and g f 1 = z ( g f n~ l + 3g f " + 2 g f n+l).

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20 Solving for / , we have

gr

= z ( 9 r - l +39fn + 2 g r ^ )

f

= r ( l + 3 / + 2 / 2)

0

=

z + ( 3 z - l ) f + 2zf2 1 -3

2

- y / l - 6 z + z* 4z

Substituting / into g — 1 -f zg T 2z g f and solving for g, we have

9

=

1

+ z9 + 2 z g f

9

~

L -z-

2zf

1_____________

1

~

o - ( 1—3z —\ / l —6z + z - \

1 - - -

V

— )

T z — v^l —6 z + z2 9 = ------------ Tz-----------1

which is the generating function for the small Schroder numbers. Therefore, by Proposition

2 .2

we can conclude that ( l^z~ ' / ^ 6z+z' .

)

js

iower tri­

angular matrix in the LDU-decomposition of S%. Similarly, so = 3 and s* = 3, A: > element ( l- 3 * -y | -

6* +.*- 1 1~->z~v2 ~ 6£±£^).

LDU-decomposition of

1

leads to the 2-admissible Riordan group This is the lower triangular matrLx of the

since g(z) = t-3*—/! - 6*-*-*:. ++ (1.3 ,1 1 ,4 5 ,1 9 7 ,...).

Now we are able to use the LDU-decomposition of 5JJ and

□

to calculate the

determinants hence the following theorem: T h e o re m 2.1. Let

and

be the Hankel matrices formed using the small Schroder

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21 numbers, then = det S* — 2 ^").

det

Furthermore, it is the unique sequence with this property. Proof: By Lemma 2.1 we know the LDU-decompositions of S'" and S^. Applying the product property of determinants, det where D = (0

= (1 ,1 ,3 ,1 1 ,4 5 ,1 9 7 ,...).

□

Having calculated the Hankel determinants for the small Schroder numbers a nat­ ural question is, "W hat are the analogous results for the closely related big Schroder and Delannoy numbers?” Propositions 2.3, 2.5 and

2 .6

answer this question.

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Table 2.3: Construction of Sequence from Hankel determinants. |/ / “| and \H„\

=

2

=

8

11 45 197

P ro p o sitio n 2.3. Let

and R„ be the Hankel matrices formed using the big

Schroder numbers, then det R„ = 2 ( 2 ), detf?* = 2 (n?‘). Proof: Using the same

technique inthe proof of Lemma 2.1, observe that when

so = 2 and s* — 3, k

> 1 in (2.1) the 2-admissible Riordan group element is

(

^ )

,

1— 3;—v/1- 6 ;

Since

= i- z- Vi-6z+z'i ^

(1 ,2 ,6 ,2 2 ,9 0 ,...) is

the generating function for the big Schroder numbers, Proposition 2.2 tells us that the lower triangular matrix of R% is ( l~z~^~p 6zt z: , f 2* i = j Furthermore, the diagonal matrix is D = (da) = < n ' . . . . Therefore. ^ 0 , i- t 3 det R^ = 2 ("). Observe that R^ = 2(S„). The effect of multiplication by a scalar, c, on the deter­

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23 minant of an n x n m atrix is multiplication by c". Therefore, det R \ = 2 "(det S \) = 2 n2 (") = 2(“^ ). We observe the determinants of |S)J| = |/2j}| =

2 (2)

□

while | 6 ’^| f- |/?^|. Consider

a 2-admissible matrix,

(

1

a i,o *4 = (aij)

1

0 2 ,0

02,1

1

0 .3 ,0

O31

03 ,2

\ On,0 On,!

0 ^ ,2 On,3 •••

1 J

9*

(

0

i = j i ^ j ‘

It also follows that the determinant of a Hankel matrix with this triple factoriztion will be det / / “ = det A • det D • det AT = 2 (2 ). Now the question becomes is there a closed form for det

= AD AT.

given

Using .4 we remove the first row and shift the rows up and form a new matrix, A = (a^j) where

= ai+l j . ^

A =

a l,0

\

1

a 2,0

a 2,l

1

1 ),

where bn satisfies the recursion

bn —Sn-ib n- i —26„_2,

6q

— 1 ) \ —«o.

Proof: Let H = A D A 7 ^ “ 1,0 “ 2,0 “3,0

\

2 -1 2 “2,l

2a.3,i

•1 22 “3,2 22

23 ■1

^ I “ 1,0 “ 2,0 “3,0 • • “n.O 1 “n, 1 “ 2,1 “3,1 • 1 “3.2 ■ • “n,2 1

“n.O \ “n - 1,0

2 “ n,l

2 2 “n,2

2 “n-rl.l

2 2a n + I ) 2

23 an,3 • • • 2 3 “n^l,3 ... 2n + 1 • 1

1

\

To get an element. h% j , in H we multiply the ith row of AD by the j th column of At . But the j th column of AT is the (j -f l) ttl row of A. Since A is 2-admissible this multiplication corresponds to the Oi+J+i o element of .4. Thus H = (hXJ) is a Hankel matrix whose (i , j ) th entry is ai+J+it0, namely Since

= A D A 7 , det

— det A ■det D • det A. We know det D =

2 (2)

and

det A = 1, so we need to compute the det A. In the same fashion as Peart in [17], we can factor A using recurrence ( 2 . 1 ) from Proposition 2.1 into A and a Stieltjes matrix S,\, A = A S a , where

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25

^ So

SA =

1

0

2

S!

1

0

2

s2

0

0

2

0

0 0

0

.

0

1

0

s3

0

...

2

>\

s„_i

By expanding the determinant of S .4 with respect to the last column we have the following: det .4 = bn where bn = s„ _ , 6 „ _ 1 - 26n_2,

60

= l , 6 i = s0. Therefore,

det H ln = 2(") • bn (n >

1 ),

where bn satisfies the recursion

hn — ^’n —15n —1 —

2h„_2,

— 1. 61 — So-

D

Taking all things together, we can state the following Theorem: T h e o re m 2.2. Given a sequence whose Hankel matrix is H = A D A T where A is a 2-admissible matrix and D = {di : ) is the diagonal matrix dij = 2 t ,i = j and dij = 0 , i ^ j , then det //° = 2(3) det

= 2^3) - 6 „

where bn satisfies the recursion bn — sn-ibn- i ~ 25„-2i

b0 — 1, b1 — Sq.

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26

Proof: The result follow from Propositions 2.2 and 2.4.

□

We can apply Theorem 2.2 to the small and big Schroder numbers with s0 = and Sk = 3, k >

1

1

and so = 2 and s* = 3, k > 1 , repectively. For the small Schroder

numbers we have, det S \ = 2^) • bn where bn = 36n_[ —26„_2, b0 = i,

=

1

which

yields bn = 1 for all n. For the big Schroder numbers we have, det R ln = 2 ^ ) -bn where bn = 36n_i —26„_2, bQ = I, b\ = 2 which yields bn = 2" for all n. Furthermore, we can compute det S'* since the det S* = 2 ^ ) . Applying Theorem 2.2, det S* =

2

^ ) •bn

where bn = 36n_i - 26n_2, b0 = 1, b^ = 3 yielding bn = 1 +- 2 F 22 F 23 F ... F 2". Now we examine the Hankel determinants of the Delannoy numbers. P ro p o sitio n 2.5. Let D\J be the Hankel matrix formed using the Delannoy num ­ bers, then det Proof: The LDU-decomposition of

= 2 n~ 12 ("). reveals the lower triangular matrix. A, is not

2-admissible but has a similar property, namely m in (m ,n ) I'm * 1"n •

^ m ,0 ® n ,0 T

^

^

—n .O

1=1 and satisfies the recurrence &n,0 = 4On_l,0 + 2an_ i i e^n,k — ®n—l,k—1 F 3

( “ ■“ )

0

Without loss of generality we extend Proposition

2.2 to accommodate this case

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27 where the diagonal matrix is

1,

D' = « , ) ==

2 *+1, 0

.

i =j =0 i= j, i >

1

*

Recurrence (2.2) implies that A — ( v 7 r v

’

V1—6 *t*~). Since 1 n i 42 y \/ 1—6 z4-2“ is

1

= A D 'A r

the generating function for the Delannoy numbers, we can conclude therefore d e tD°n = 2 "_ l 2 (").

□

In the case D„, the LDU-decomposition revealed the lower triangular matrix involved fractions. D\ is shown in Figure 2.4.

(

13 63 321 \ 13 63 321 1683 63 321 1683 8989 V 321 1683 8989 48639 /

D\

/

\

( 3 -2 0

3

0 f • 22

1

0

0

0

12 3

1 36 s; 219 5

0

0

1 91 9

0

0

0

1

0

0

21

\ 107

\

\

0

0

0 9 ^3 5 *“ 0

0

0

1

0 17 9 4 9 ‘ "

0

12 3

/ 1

/

107

0

21 36 5 1

0

0

1

219 5 21

ij

Figure 2.4: LDU-decomposition of D\

Even though Riordan group elements are defined over the complex numbers, attem pts to find the dot diagram for f ( z ) were unsuccessful. In the spirit of Theorem 2.2, we prove the following proposition for D\.

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28 P ro p o sitio n 2.6. Let D\ be the Hankel matrix formed using the Delannoy num ­ bers, then det D ln = 2 " - 12(")(l + 2n). Proof: Let A = (atJ) be the lower triangular matrix from the LDU-decomposition of D”. Let A = (Uij) where ah: = Oi+1j , we have D ln = AD 'AT. Figure 2.5 gives an example, D\.

/

13 63 \ 321

D\

f

3 13 63

1

t

0 6

1

33

9

0 0

\

13 321 \ 63 63 321 1683 321 1683 8989 1683 8989 48639 J

( 1 0

0 4

1

0

12

^ 0 0

/

0

\

(

0

0

1

0

0

0

8

0

0

0

16

3 13 63 \ 1 6 33 0 1 9 0

0

1

/

Figure 2.5: Decomposition of D\ = A t f AT

Applying the product property of determinants we have,

det D xn = det A ■det D' • det AT.

As in the proof of Proposition 2.4, we define a recurrence relationship for det A = bn. In this case, the recurrence bn = 36n_i —26n_2, bn =

1

5i = 3,

62

= 5 which yields

+ 2 " Therefore,

d e tD ‘ = 2 " - 12 (”) (l + 2 ").

□

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29

2.3

Combinatorial Interpretations

The following describes the combinatorial interpretations of the 2-admissible matrix in Figure 2.6. /

\

1 1

1

3

4 1 17 7 45 76 40 11

V

1 10

w

Figure 2.6: Example of 2-admissible matrix. Ls

The (i , j ) th entries correspond to the paths above the x-axis of length i that end at height j consisting of up, level and down steps, i. e., {( 1 . 1 ), ( 1 , 0 ), ( 1 , —1 )}, respectively, with the following restrictions: (i) Up steps have one color; (ii) Level steps have one color at height zero and three colors otherwise; and, (iii) Down steps always have two colors. Consider the entries in the second row, r 2 = 3 4 1 , Figure 2.7 illustrates the paths described above.

R-Red G-Green B-Blue Figure 2.7: Illustration of second row of Ls

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30 Examining the row multiplication, rm * rn, we combine the paths in each row to make new paths. Therefore, a combinatorial interpretation of a 2-admissible matrix is: Given the paths in r m of length m and the paths in r n of length n which end at the same height, say k, will yield a path of length m + n at height zero by reversing the second path to start at k and thus ending at height zero. By reversing the second path each up step turns into a down step which results in a multiple of 2* at each height k = 0 .1 .2 ,3 ..... Figure 2.8 shows the 11 paths from n * r 3 = 1 * 2 -F 1 * 4 * 2 = 11 = a:{0-

R-Red G-Green B-Blue T[ - solid paths and r 2 - dotted paths Figure 2 .8 : Illustration of row multiplication of r[ * r 2 in Ls

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Chapter 3 Generalization of Schroder Num bers Results In this chapter, we define a class of sequences with detH® = detH„ =

We

present the main result and examine the case of m = 4 relative to its combinatorial significance.

3.1

m-admissible

Motivated by Aigner’s definition of an admissible matrix in [1] and the LDUdecomposition of the small Schroder numbers yielding a 2-admissible matrix, we generalize the definition to m-admissible. D efinition 3.1. A lower triangular matrix, .4, with main diagonal entries equal to mi n ( i . j )

I, is m -adm issible if r, * rj = a1+j 0 for all i , j where r, * r, :=

^ at kahkmk k=o

and Ti = (a^o, aj.i, a*,2 , •••) denotes the ith row of A. We extend the results from Chapter 2 to the m-admissible case. This gives us the following propositions: 31

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32 Proposition 3.1. Let A = (an,*) be a matrix that satisfies the following recursion 7i,k

On—i,fc—i ~t~SkQn—i.k ~F man—itk+ i • ^ ^

1

(3.1) ao,o = 1,

where

a 0,jt = 0 f o r k > 0

= bn fo r all n and s0 = b0, s t = 6 , - 60, • • • , sn =b n - b n_ l . Then A is

a m-admissible matrix. Proof: We can just replace

2

with m in the proof of Proposition

2. 1

and the result

follows.

□

Proposition 3.2. If a Riordan group matrix, A = (g, f ) is m-admissible then H = .4Di4r is a Hankel matrix whose (i . j ) th entry is alrj(h the coefficients of g(z), where D = {dtJ) is the diagonal matrix (l.rnz), i.e., dtJ - m l, i — j and dij = 0, i

j.

Proof: We can ju st replace 2 with m in the proof of Proposition 2.2 and the result follows.

□

Proposition 3.3. Given a sequence whose Hankel matrix is / / “ = A D A T where A is a m-admissible matrix and D = (dtJ) is the diagonal matrix di } = m l. i = j and ditj = 0 , i / j , then det det

= m (2) = mW • bn

where bn satisfies the recursion bn

Sn-l&n-l

< m6n_2 ,

6q

— li b\ — S q .

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33 Proof: We can just replace 2 with rn in the proof of Theorem 2.2 and the result follows.

□

If the sequence in Proposition 3.3 yields d e t//° = det H\ =

we call it

the m -adm issible sequence. In general, the lower triangular m atrix from the decomposition of H® formed with the m-admissible sequence is a Riordan group m atrix that has the dot diagram shown in Figure 3.1.

1

m

1

rn

1 rn

V '* *N J/ Figure 3.1: Dot Diagram for matrix yielded from rn-admissible sequence

Table 3.1 shows the first few terms of some of the m-admissible sequences for m — 1,2,3,4,5 and their associated reference if applicable in Sloane's On-line Ency­ clopedia of Integer Sequences [27]. Notice when m =

1

the sequence is the Catalan

numbers and when m — 2 the sequence is the small Schroder numbers.

3.2

4-admissible Sequence

Upon extending the results of the 2-admissible matrix, which is associated with r a

= w i \ = 2

1}

which can also be viewed as a problem involving bishop moves.

This connection motivated two of the four lattice path settings presented in this

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35 research, Free Bishop Moves (Sec. 3.2.3) and Up Bishop Moves (Sec. 3.2.4). Two additional settings, 4-admissible Translation (Sec. 3.2.1) and Schroder Analog (Sec. 3.2.2), turn out to give the same sequence.

3.2.1

4- admissible Translation

The A-admissible T ra n s la tio n paths are lattice paths from (0,0) to (rc, 0) not going below the x-axis, using the step set {(1,1), (1, —1), (1.0)} where there are 4 types of down steps, 5 types of level steps above height zero, 1 type of level step at height zero and 1 type of up step. This setting is motivated by the 4-admissible translation of the LDU-decomposition of a Hankel matrix with detH “ = detH* — 4(2). The A-admissible translation matrix has the dot diagram in Figure 3.1 with rn = 4 which implies the generating function for this setting is

l ^ :iz ~ >/^

1(>z" 9z2

1.1.5,29,185,.... Figure 3.2 illustrates the lattice paths of the first terms.

r 1 1 1 1 1 1 1 1

Figure 3.2: A-admissible Translation Paths

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36

3.2.2

Schroder Analog

The 4-admissible Schroder A nalog paths go from (0,0) to (2n, 0) not going below the x-axis using the step set {( 1 , 1 ), ( 1 . —1 ). (2 , 0 )} where there are no level steps at height zero and three types of level steps above height zero. This setting is denoted the Schroder Analog because it is motivated by the random walk enumera­ tion for the small Schroder numbers described in Section 2 . 1 . Using the pictures in Figure 3.3, we derive the generating function, g{z), for the 4,-admissible Schroder Analog.

Fig.3.3.A: g{z) = 1 + z2g(z)G(z)

3z2 f G U ) \ Fig.3.3.B: G(z) = l+ z 2G2{z)+3z2G{z) Figure 3.3: Schroder Analog Picture Representation of g(z)

Consider the first return to the x-axis, Figure 3.3.A shows g(z) will satisfy

g{z) = 1 + z2G(z)g{z)

(3.2)

where G( z ) is the generating function for the A-admissible Schroder Analog paths that allow level steps at height zero. To find the generating function G(z), Fig­

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37 ure 3.3. B shows that G(z) will satisfy

G(z) =

1

+ z 2G( z ) + 3z 2G{z)

which implies . CM

1 - 3z2 - v/1 - IO2 2 + 9z* --------------- £ 3 ----------------■

Substituting G(z) into (3.2) and solving for g(z) we find

, x 1 + 3 z 2 - y/1 - 1 0 z 2 + 9z4 g(z) = ------------- 1 .0 ,1 ,0 .5 .0 ,2 9 ,0 .1 8 5 ____________________ Sz Figure 3.4 shows the first few terms for the Schroder Analog lattice paths.

l / \ l 3 1 /

\9 X

\6 9 185 185

Figure 3.4: 4-admissible Schroder Analog Lattice Path Enumeration

3.2.3

Free Bishop M oves

The i-adm issible Free B ishop M oves paths are lattice paths from (0,0) to (2n. 0) not going below the x-axis using the step set {(fc, k), (k , —k ) , k > 1}. Notice all steps can be of any length. This setting is motivated by Woan’s [32] solution to the following problem:

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38 “How many lattice paths are there from (0,0) to (2n,0) not going below the x-axis using the steps {(/:, ± k ) , k >

1 }?”

Since it may also be viewed as a problem involving bishop moves in chess, it is referred to here as Free Bishop Moves. Woan derived the following recursive relationship depicted in Figure 3.5 for the generating function, g(z), by considering the first return to the x-axis: g(z) =

1

+ 4z 2g2(z) - 3z 2 g(z).

4 choices

over counted 3 tim es

R or G

R or G

x

1

-f

4 z2g2(z)

-

3z2g(z)

Figure 3.5: Free Bishop Moves Picture Representation of g(z)

The first term, 1, counts the em pty path. Given a random path, we construct the recurrence relation of the path based on its first return to the x-axis. The path up until the first return is a p ath whose first step is an up step of length one or k and whose last step is a down step of length one or k. Each intermediate point on the run of up steps can be viewed as G (go) or R (stop). We can thus proceed one unit step at a time. At the end of the first unit up step, either there is a G or R indicator. We can treat the last unit down step before the first return similarly with a G or R indicator. Marking the first unit up and last unit down step with

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2

39 and the remaining part of the path with g(z), we have the term 4z2g2(z). For these paths, if the g(z) term before the first return is the empty path, i.e.. [z°]^(r) =

1.

we have the term \ z 2g{z). This means we have counted the paths th at begin with UP, DOWN four times. Thus we subtract 3z 2g(z). From the recursion we have

9(z)

i + 3z2- v/T^IoPTgl7

1}

where there are no

level steps at height zero and two types of level steps above height zero. Notice the up steps can be of any length. This setting is a variation of the EYee Bishop Moves setting mentioned in the previous section. Figure 3.7 is an enumeration array for the Up Bishop Moves lattice paths. We notice that if we rotate the array, we will need to eliminate the bottom row to have a Riordan group element.

8 36 98 185 185

Figure 3.7: Up Bishop Moves Lattice P ath Enumeration

So we consider an equivalent problem, which we will call Down Bishop Moves. If we consider the paths that go from (0,0) to (2n, 0) not going below the x-axis using the step set {( 1 , 1 ), (fc, —k), (2 , 0 ), k >

1}

where there are no level steps at

height zero and two types of level steps above height zero. Figure 3.8 shows the development of this setting. We notice we have a Riordan matrix without eliminating the bottom row. Ro-

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41

46 128 185

Figure 3.8: Down Bishop Moves Lattice P ath Enumeration tating this array, we consider this Riordan matrix, E, and find (g ( z ) , f ( z )). (

E

\

1 0

1

1

0

1

0

4

0

1

5

0

7

0

0

21

0

10

0

1 1

29

0

46

0

13

0

1

0

128

0

80

0

16

0

1

185

0

314

0

123

0

20

0

\

• /

Using the formation of the array, we can construct the dot diagrams for E. Figure 3.9 shows the dot diagrams for g(z) and f (z ) . Using the dot diagram for / we have,

gfh

z g f k~i + 2 z 2f k + g ( z f k+l + z 2f k+2 + z zf k+i + ...)

fk f

)■

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42

1

1

io

io

1 O

2

io

1 O

io io

X O

0 X 0

•

•

9

f

Figure 3.9: Dot Diagram for Down Bishop Moves M atrix Solving for / , we have

f{z) =

1

+/j

1 0 ^ 2—

4 Z'j — 4 z

o, 1.0,1.0,13.0, 71.0.441.

From the clot diagram of g, we have

9

—

f + ( z g f -f z 2f 2g + z 3f 3g 4- z ^ f ^ g 4- . . . )

9 = 1+9

(t^ )

l - z f

9

=

1 - 2 zf

Substituting / into g, it follows that

9 {z) =

1 + 3 2 2

VL 2 i0z2 + 9z4 4-)- 1,0,1.0,5,0,29,0,185, oz

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43

3.3

Bijective proofs

In the last section we proved that each of the four settings were counted by the sequence (1 ,1,5,29,185,...). The obvious next step is to seek bijective proofs. In general, finding a bijective proof can be very difficult. This section provides bijective proofs among the four settings, one of which was adapted from other work [13]. Using the Schroder Analog as the focal point, the scheme shown in Figure 3.10 will be used to establish the bijections. In showing the correspondence between the Schroder Analog and the Free Bishop Moves (Sec. 3.3.2), Lalanne’s presentation in [13j of an involution due to Kreweras was used as a basis for the bijections. 4-admissible Translation Free Bishop Moves

z Schroder Analog

Up Bishop Moves

Figure 3.10: Schematic for Bijective Proofs

3.3.1

4-admissible Translation ■• Schroder A nalog

The ‘l-admissible Translation paths are length of n while the Schroder Analog paths are of length 2n. To adjust the 4-admissible Translations paths we double the size of each step, hence making the paths length 2n. The step set becomes {(2,2), (2 ,—2), (2,0)} up, down and level, respectively. We also adopt the conven­ tion of associating the different colors with a number. To form these paths, double

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44 the size of each step, label the middle of each step by the number corresponding to the color of the step. Level steps are labeled 1 through 5 and down steps are labeled 1 through 4. Since level steps at height zero and up steps have one color choice, they will not be labeled. We call these paths 4-admissible Translation*. (=s>) Given a 4-admissible Translation* path, P, perform the following: 1. Construct path P' by converting level steps in P. (a) Level steps labeled 1-3 remain level steps with corresponding colors 1-3. (b) Turn level steps labeled 4 into valleys (down-up step) with no label. (c) Turn level steps labeled 5 into hills (up-down step) with no label. (d) Turn level steps at height zero into hills (up-down step) with no label. 2. Construct path P" by converting down steps in P'. (a) Down steps labeled 4 remain down steps with no label. (b) Convert down steps labeled 1-3 to corresponding level steps as follows: i. From the middle of the each down step, draw a horizontal line from east to west. The first vertex this line intersects becomes the start of a level step formed from the up step above this point and the down step above the point of origin; and, ii. Label the inserted level step with the label of the original down step. 3. Up steps of the form {(2,2)} become two up steps of the form {(1,1)}. Like­ wise, down steps of the form {( 2 , —2 )} become two down steps of the form

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45

Path P" is the Schroder Analog path corresponding to the given -{-admissible Trans­ lation* path P. 5

Figure 3.11: Illustration of A-admissible Translation* to Schroder Analog

To convert the Schroder Analog paths into A-admissible Translation* paths, we pro­ ceed from left to right considering two units at a time. Convert the pairs of steps as described below, where we denote the first half of a level step by Level, while LEVEL denotes the full step. () Given a Free Bishop Move path, R, perform the following: 1. Label stopping points on the up steps with 1 and stopping points on the down steps w ith

2

.

2. For each step of length m > 2 there are m -

1

double rises. For each counterstep

of length n > 2 there are n — 1 double descents. From the first double rise, draw a —45° line and from the first double descent draw a 45° line until they intersect. If the double rise and/or double descent has been marked with a number, transfer the marking to the intersection point.

Continue drawing

a —45° line from the next double rise and a 45° line from the next double

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48

Q'

Q"": Figure 3.12: Illustration of Schroder Analog to 4-admissible Translation* descent, maintaining markings as before mentioned in this manner until all possibilities axe exhausted. 3. Construct a new path, R! under the original path by using the intersection points as the vertices for the new path. Convert up/down steps labeled 1, 2

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49 or 1/2 to level steps as follows: For vertices labeled 1, convert this step into a level step labeled 1, likewise for vertices labeled with 2. For vertices marked with 1/2 this step becomes a level step colored 3. 4. Rotate path 180°. P ath R! is the Schroder Analog path corresponding to the given Free Bishop Move path R.

R :

R!: Figure 3.13: Illustration of Free Bishop Moves to Schroder Analog

( 2, is composed of k unit sized up steps. Between the stopping points the vertices of the unit steps are called intermediate points. For each up step with length k > 2, there are k —L intermediate points. The main difference between Up Bishop Moves paths and Schroder Analog paths is that the Up Bishop Moves paths have possible intermediate points on up steps and 2 types of level steps while the Schroder Analog has 3 types of level steps.

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52 We base the bijection on finding a correspondence between the intermediate points in the Up Bishop Moves paths to the level steps labeled 3 in the Schroder Analog. If there are no up steps with intermediate points on an Up Bishop Moves path, then the path has an identical representation in the Schroder Analog. Likewise, if there are no level steps labeled 3 in a Schroder Analog path, then its corresponding Up Bishop Moves path is identical. (=>) Given an Up Bishop Moves path. S, perform the following:

1

. If S contains no intermediate points, then the corresponding Schroder Analog path is S.

2

. If S contains intermediate points, then we convert them as follows: (a) Mark all intermediate points with an x. Consider each of these k — 1 intermediate points between the stopping points. (b) Moving from left to right along the path, start at the first intermediate point. Draw a horizontal line from west to east until it intersects the “end” of a down step. This will be the place to insert a level step labeled 3 using the unit UP step above the intermediate point and the DOWN step above the intersection point to make up the LEVEL step. Remove this intermediate point marking. Reconstruct this new path, call it S'. (c) If there is an intermediate point on S', repeat Step 2a, reconstruct this new path, call it S".

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53 (d) Repeat this process n times.

will not contain any interm ediate

points. In other words all up steps are of length 1. P ath S (n) is the Schroder Analog path corresponding to the given Up Bishop Moves path S.

S:

2

3

Figure 3.15: Illustration of Up Bishop Moves to Schroder Analog

( 2. Reconstruct this new path, call it Q '. (b) If there is a level step labeled 3 on Q', repeat Step 2a, reconstruct this new path, call it Q". (c) Repeat this process n times.

Q^n) will not contain any level steps of

length 3. Path

is the Up Bishop Moves path corresponding to the given Schroder Analog

path Q.

1

X

—2

Figure 3.16: Illustration of Schroder Analog to Up Bishop Moves

Table 3.2 illustrates the bijections for each lattice path setting for [zs\g(z) = 29.

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55

Table 3.2: Bijective Correspondence for [z6\g(z) = 29 A-admissible Translation*

Schroder Analog

Free Bishop Moves

Up Bishop Moves

CONTINUED

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56 Table 3.2 (continued) 4-admissible Translation*

3

Schroder Analog

Free Bishop Moves

Up Bishop Moves

3

A

A

A

1 aA

3 z \A x CONTINUED

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57 Table 3.2 (continued) A-admissible Translation*

Schroder Analog

Free Bishop Moves

Up Bishop Moves

CONTINUED

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58 Table 3.2 (continued) A-admissible Translation*

Schroder Analog

Free Bishop Moves

Up Bishop Moves

2

\ 3

\ 3.4

M ain R esult

We can now generalize from 2-admissible and A-admissible to m-admissible at this point. Without more than slight modifications in the proofs, the following theorems hold. Theorem 3.1. The m -adm issible sequence has generating function. 1 -f (m — i )z — y j 1 — (2m 4- 2)z + (m — l)2z2 2 mz It is the unique sequence whose Hankel determinants are detH® = detH* = mb').

□

Theorem 3.2. Given a positive integer, m , each setting below yields either the madmissible sequence or the aerated m-admissible sequence if defined. When m > A each setting is defined. A. m — a d m issib le Translation: Lattice paths from (0.0) to (n, 0 ) not going below x-axis using the step set {(1,1), (1, —1), (1,0)} where there are m types

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of down steps, m + I types of level steps above height zero, 1 type o f level step at height zero and 1 type of up step, m > 1.

B. Schroder Analog: Lattice paths from (0,0) to (2 n, 0 ) not going below the x-axis using the step set {( 1 , 1 ), ( 1 , —1 ), (2 , 0 )} where there are no level steps at height zero and m — 1 types o f level steps above height zero, m > 1.

C. Free Bishop Moves: Lattice paths from (0,0) to (2n.0) not going below the x-axis using the step set {(fc, k), (k , —k), (2k, 0 ), k > 1} where there are no level steps at height zero and m —4 types of level steps above height zero, m > 4. D. Up Bishop Moves: Lattice paths from (0,0) to (2n, 0 ) not going below the x-axis using the step set {(k , k ), (1, —L), (2 ,0 ),fc > 1} where there are no level steps at height zero andm —2 types of level steps above height zero, rn > 2.

3.5

□

A dditional Combinatorial Interpretations

Within the lattice path setting, we can consider the big Schroder Analog of the m-admissible sequence. As with the small and big Schroder numbers, we define the big Schroder Analog of the m-admissible sequence the same as the Schroder Analog with level steps allowed at height zero. This perturbation of the Schroder Analog gives us the generating function for the big Schroder Analog which is .

1

—(m —1 ) 2 — y / l — ( 2 m 4- 2 )z + (m — 1 )2z 2

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60 For fixed value of m , we see that the big Schroder Analog is essentially numerically rri times the Schroder Analog. In addition to the lattice path settings described in the previous section, we present two additional combinatorial settings for m-admissible sequences. In Chap­ ter 2 . we introduced State Diagram Enumeration and Parallelogram Polyominoes. Each of these settings can be extended to the m-admissible case as described below.

S ta te Diagram Enum eration. Consider starting at a state zero and being able to move to the right using one kind of step each time and returning to the state using either one way from odd positions and m ways from even position when going left as depicted in Figure 3.17. The m-admissible sequences count the number of ways to leave and return to the initial state zero under these circumstances.

m ways rn ways Figure 3.17: m-admissible State Diagram Enumeration

Parallelogram Polyom inoes. Parallelogram Polyominoes with white and black columns are called zebras. Consider parallelogram polyominoes whose columns can be any of m colors, the big Schroder numbers Analog to the m-admissible sequences count the number of such parallelogram polyominoes with semi­ perimeter n. In [31], Sulanke provides three recurrence relationships for paral-

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61 lelogram polyominoes. In particular one of the three recurrence relationships presented leads to this interpretation of the m-admissible big Schroder num­ bers Analog relationship to parallelogram polyominoes.

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Chapter 4 Disjoint M otzkin Path System s In this chapter, we present preliminary results for Disjoint Path Systems formed using Motzkin paths. The results are motivated by a well known result for the combinatorial relationship between Hankel matrices and disjoint path systems.

4.1

D isjoint paths and Hankel D eterm inants

In UA determ inant property of Catalan Numbers” [15], Mays and Wojciechowski discuss the following theorem from a paper by Gronau. et al. [10]. The theorem also has previous roots in papers by Gessel-Viennot [7] and Lindstrom [14]. Theorem 4.1 (G ronau, et al. [10]). Letpij be the number of paths leading from at to bj in G, let p + be the number of disjoint path systems W in (G. .4, .4') for which (t {W) is an even permutation, and let p~ be the number of such systems fo r which cr{W) is odd. Then det(p.y) = p+ —p~. This theorem relates detH\. to the number of pairwise vertex disjoint path sys­ tems. Essentially, the difference in the even and odd permutations is the Hankel 62

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63 determinant of H i, where k corresponds to the number of paths in the disjoint path system and t determines the initial distance between ax and a\. Also in [15], the results for the Catalan numbers include proving the total num­ ber of vertex disjoint path systems is the Hankel determinant of Cl by showing there are no disjoint path systems th at are odd permutations. This research is motivated by the question, “W hat is the total number of disjoint path systems for other paths which do have odd permutations, such as, Motzkin paths?”

4.2

M otzkin Paths

Given the Motzkin numbers with generating function, , 1 - 2 - \ / l — 2z — 3z* M (z) = ----------- ^ — 5--------------^ 1 ,1 , 2 ,4 , 9 ,2 1 . .. , Az* Ml denotes the k by k Hankel matrix starting with the tth Motzkin number. Using the Motzkin paths to find the total number of disjoint path systems, Table 4.1 is a list of hand computed results. Based on these computations the results in this section were discovered. For any vertex disjoint Motzkin path system with k paths and initial points ax and a'j, t units apart, the steps prior to the vertex whose abscissa is the same as the abscissa of a x are up steps. Similarly, the steps after the vertex whose abscissa is the same as the abscissa of a' are down steps. So it is equivalent to consider the “interior” of the path system to count the total number of vertex disjoint path systems. Figure 4.1 shows the “interior” of the Motzkin disjoint path systems.

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64

Table 4.1: Motzkin Path Results det Mfc P+ +P

P

P+ ~ P~

1 1 1

0 0 0

1 1 1

1 1 1

1 1 1

0 1

1 0 -1 -1 0

1

8

6

1

13

2

0

2

2

4 9

2

2

6

3 4 4 5

15 40 104 273

II p+ t= 0 k = 1 . This relationship is equivalent to the Fibonacci recurrence. Therefore, the total number of disjoint Motzkin path systems with k paths when t =

1

is the k tb

Fibonacci number as defined above.

I*

* 1'

Pi = 1

P2 =

2

Pit = Pfc-l + Pfc- 2 Figure 4.3: t =

1

Disjoint Motzkin Path Systems

□

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67 P ro p o sitio n 4.3. When t = 2, the total number of disjoint path systems with k paths is the product of the Fibonacci numbers, Fk and f t+ 1 , where the Fibonacci numbers are defined by Fn = f t - i -f f t _ 2, f t — 1 , Fy = 1 . Proof: Since t = 2, a\ and a\ are two units apart. We provide a proof by construction using Figure 4.4.

X 3

.

.

.

3'

2 . 1 .

.

.

.

.

2' 1'

t

t

ft

ft

k

.

.

kf

2

...

1

.

.

t Fk

2

1

.

1'

.

.

'

-f 6 = 15

k

S

2

2

... '

1'

1

'

.

. . . Pi

ft =

.

2

. \

k'

2

.

.

.

1'

t Fk

Pk-i Pk = F t + P t-i

Figure 4.4: t = 2 Disjoint Motzkin Path Systems

When k = 1, the total number of systems is 2. The path systems fall into two

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68 categories, either the path system uses the top vertex or it does not. In other words either the path from k to