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HOW ARD UN IVER SITY
Random Walks, Trees and Extensions
of
Riordan Group Techniques
A Dissertation
Submitted to the Faculty of the
Graduate School
of
HOW ARD U N IVER SITY
in partial fulfillment of
the requirements for the
degree
DOCTOR OF PHILOSOPHY
Department of Mathematics
by
Naiom i Tuere Cameron
Washington, D.C.
May 2002
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UMI Number: 3066485
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HOW ARD UN IVER SITY
GRADUATE SCHOOL
DEPARTM ENT OF M ATHEM ATICS
DISSERTATION COMMITTEE
Neil Hindman, Ph.D.
Chairperson
Paul Peart, Ph.D.
Tjmoi %{ j^xLpiy-u^
Louis W. Shapiro, Ph.D.
Wen-Jin Woan, Ph.D.
^ohWoodson, Ph.D
Associate Professor of Mathematics
Morgan State University
Louis W. Shapiro, Ph.D.
Dissertation Advisor
Candidate: Naiomi Tuere Cameron
Date of Defense: January 17, 2002
ii
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Dedication
Dedicated to my beloved brother,
Jarrell Ali Cameron
(December 5, 1977 - September 14, 2001)
You will live forever in my heart.
iii
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Acknowledgments
While it is not possible to name every person who supported me during my
graduate studies, I would like to expressly thank those individuals whose support
proved critical to the achievement of my goals.
First and foremost, I thank my Lord and Saviour, Jesus Christ, through whom
all things are possible.
I would like to express my most sincerest appreciation to my beloved family,
Jason, Kariyma and Omar, for their unconditional love and support throughout
this chapter of our life. I could not have done this without their support. I also
want to thank my Mom, my Dad, my sister Njeri and my brother Jawanza for their
consistent love and understanding. I must also thank my mother-in-law Patricia
Murphy for her constant encouragement.
My advisor, Louis Shapiro, deserves the credit for raising many of the questions
addressed in this research, and I would like to thank him for his care and guidance
throughout this project. I must also thank the members of my committee and the
Combinatorics Seminar, Paul Peart, Wen-Jin Woan, Seyoum Getu, Leon Woodson,
Neil Hindman and Emeric Deutsch, for their invaluable input. In addition, the
members of the Mathematics Department at Howard University have provided im
mense support and encouragement. In particular, I would like to mention Adeniran
Adeboye, Richard Bourgin, Francois Ramaroson, the graduate student body and
the administrative support staff. I must extend a special thanks to Cora Sadosky
for her support and mentorship to me as a mathematician and as a woman.
iv
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There are many extended family members and friends whose support of me and
my family have been crucial to the completion of my dissertation. They include
Lisa and Rick Square, Karen McFadden, Carla Jones-Chew, Sherry Scott, Latifa
Barnett, Deangela Hill, Rena Stevens, Tiandra Speaks, Kim Cameron-Dominguez
and all of my other family members, both big and small.
I am especially grateful to two young women mathematicians whose friendship I
treasure: Jillian McLeod and Lynnell Matthews. Jillian, your friendship has made
my life better. Thank you for your consistent care and attention to me. Lynnell,
this dissertation is not complete without yours. Thank you for the many days when
you provided an ear for mathematics and the countless times when you provided an
ear for a friend.
v
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Abstract
The Catalan number sequence
1,1,2,5,14,42,.
n + l \ nJ
is one of the most important sequences in all of enumerative combinatorics. Richard
Stanley cites at least 66 combinatorial settings where the sequence appears [7].
Among the numerous interpretations of the Catalan numbers, we have the number
of paths restricted to the first quadrant of the x1 y-plane starting a.t (0, 0) and ending
at (2n,0) using “up” steps (1,1) or “down” steps (1,-1) (known as Dyck paths).
This research will focus on a generalization of the Catalan numbers which can be
interpreted as taking Dyck paths and perturbing the length of the down step. One
particular example of this generalization is given by the sequence of numbers known
as the ternary numbers,
1,1,3,12,55,273,1428,7752,43263,..., — ( in) ,....
2n + 1 \ n J
This sequence arises in many natural contexts and extensions of known results re
lated to combinatorial objects such as paths, trees, permutations, partitions, Young
tableaux and dissections of convex polygons. Hence, we use this sequence as an
important example of something which lies on the boundary of what is known and
what is new. Much of this research is an attempt to extend many of the known
results for the Catalan numbers to ternary and m-ary numbers.
vi
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In this dissertation, we establish, in the setting of the ternary numbers, several
analogues to the better known Catalan numbers setting. We present an analogue
to the Chung-Feller Theorem which says that for paths from (0,0) to (3n, 0) with
step set {(1,1), (1, —2)}, the number of up (1,1) steps above the rr-axis is uniformly
distributed. We also present analogues to the Binomial, Motzkin and Fine gener
ating functions and discuss combinatorial interpretations of each. In addition, we
establish some computational results regarding the area bounded by ternary paths
and the number of returns to the ar-axis. We also present generating function proofs
for the area under Dyck and ternary paths, an interesting connection to weighted
trees, and a conjecture for a closed form generating function for the area under
ternary paths.
vn
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