Table Of ContentDETERMINANTS, PERMANENTS, AND
THE ENUMERATION OF
FOREST-PARTITIONS
By
Adam H. Berliner
A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
at the
UNIVERSITY OF WISCONSIN - MADISON
2009
UMI Number: 3384476
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A dissertation entitled
Determinants, Permanents, and the
Enumeration of Forest-partitions
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Adam H. Berliner
Date of Final Oral Examination: July 10, 2009
Month & Year Degree to be awarded: December May August 2009
Approval Signatures of Dissertation Committee
Crac:-
(STUCA^IUL
U/JaL
Signature, Dean of Graduate School
i
Abstract
First, we observe that one can regard the domain of the classical partition function p(n)
to be the paths P on n vertices. Replacing P with an arbitrary tree T on n vertices
n n n
(or any graph G on n vertices), we get a more general enumerative function that we
n
call a forest-partition function. We evaluate this function for certain classes of trees,
and also obtain some basic properties.
Zeilberger has given a combinatorial proof of Dodgson's rule for calculating deter
minants. By use of Muirs law of extensible minors, Dodgson's rule can be generalized
to an identity we call the Dodgson/Muir identity. We give a combinatorial proof of this
identity which extends Zeilberger's proof.
In the final chapter, we observe how a problem of Polya leads naturally to the study
of SNS and convertible matrices. A convertible matrix allows one to compute the per
manent (a notoriously difficult computational problem) by a determinant (a computa
tionally easy problem) by using a signing of the matrix. We then construct a sequence
of maximal convertible matrices which achieve a new bound for the number of nonzero
entries relative to the size of the matrix. Finally, using more than one matrix to con
vert the permanent problem into one involving determinants, we discuss the notion of
m-convertibility. Then, relaxing the requirement that we look at signings of the original
matrix, we investigate the example of J and convert its permanent in a more general
n
way.
ii
Acknowledgements
This thesis is dedicated to my parents Karol and Anthony and my brother Daniel, who
have been there for me every step of the way. Thank you for your interest in everything
I have done and for your enthusiasm. I could not have gotten this far without your
support and I know I will certainly put that to good use in the future, as well. I would
also like to thank several others who have been instrumental in my success.
First and foremost I would like to thank my advisor, Richard A. Brualdi, for his
incredible wealth of knowledge, patience, and enthusiasm. I very much appreciate your
guidance and the opportunities you have given me. Additionally, I am fortunate to have
been able to work with Louis Deaett, Ulrike Bostelmann Loskamp, Kathleen Kiernan,
Michael Schroeder, Seth Meyer, and Dimitrije Kostic along the way. Furthermore, I
would like to thank the University of Wisconsin Mathematics Department and the Na
tional Science Foundation for supporting me with funding through the VIGRE program.
I would also like to thank Robert L. Wilson, Concetta Gomez, the Wisconsin Emerging
Scholars Program, and all of my former students for helping me become the teacher I am
today and for helping me develop valuable tools which I know will assist me in my career
as an educator. Additionally, I would like to thank Dan Butler at Mounds View High
School, who in middle and high school helped me develop my interest in mathematics,
and the faculty of Carleton College, who did an amazing job preparing me for graduate
school and enhancing my excitement for teaching mathematics. Finally, thank you to
all of my friends and colleagues in Madison for creating such a positive and supportive
environment for my studies. I could not have done it without all of you!
iii
Contents
Abstract i
Acknowledgements ii
1 Overview 1
1.1 Enumeration of Forest-partitions 1
1.2 Determinants and Permanents 2
2 Forest-Partitions 5
2.1 Integer Partitions and Graphs 5
2.2 Enumeration of Forest-Partitions 11
3 The Dodgson/Muir Determinantal Identity 26
3.1 Dodgson's Rule and Muir's Law 26
3.2 Combinatorial Proof of the Dodgson/Muir Identity 29
4 Calculating Permanents via Determinants 38
4.1 Convertible Matrices 38
4.2 m-Convertible Matrices 45
4.3 General m-Convertibility and J 52
n
Bibliography 68
1
Chapter 1
Overview
In this chapter, we give an overview of Chapters 2, 3, and 4. In doing this, we will give
some historical perspective from which we approach the topics covered in those chapters.
1.1 Enumeration of Forest-partitions
Integer partitions are one of the most widely-studied objects in number theory and
combinatorics. In 1918, Hardy and Ramanujan found an asymptotic formula for the
number p(n) of partitions of an integer n, namely
exp(7r^2n/3)
In 1937, Rademacher followed this up with an exact formula, however it is quite difficult
to compute. Other work has centered around proving partition congruences as well
as finding combinatorial proofs for partition function-related theorems. The books [1]
and [2] contain an advanced and elementary, respectively, exposition of the theory of
partitions.
In Chapter 2, we connect the integer partition function to graphs (specifically, trees).
If, for each partition TT of n, the n vertices of a graph can be partitioned into sets whose
sizes correspond to the parts of TV such that each set of the partition induces a connected
subgraph, then the graph is arbitrarily vertex decomposable (avd). There are many
2
results in the literature concerning avd graphs; see [3, 4, 16, 17, 18, 29] and other
references contained therein. We define a new partition function for each sequence of
graphs. In particular, the sequence of paths on n vertices yields the integer partition
function. We then calculate this function (called the forest-partition function) for some
sequences of trees which closely resemble paths and stars. For some of these, we use
(1.1) and other facts about p(n) to compare the growth of the new partition functions
with that of the classical integer partition. In particular, we show that if a tree has
a bounded radius and very few vertices of unbounded degree, then the forest-partition
function is bounded by a polynomial in the number of vertices.
1.2 Determinants and Permanents
Although a theorem may already be known to be true, mathematicians are constantly
seeking new explanations as to why. In particular, combinatorial (or bijective) proofs
are generally more illuminating and are thus highly sought-after. Some of the most
interesting and important theorems of linear algebra and combinatorics have yielded
combinatorial proofs. For example, [34] gives a combinatorial proof of the famous Cayley-
Hamilton Theorem, which states that a matrix satisfies its own characteristic polynomial.
This, in turn, leads to a largely graph-theoretic proof of the existence of the Jordan
Canonical Form of a matrix (see [8]). Several identities involving determinants of a
matrix also have combinatorial explanations.
Dodgson's Rule is one such identity, and a combinatorial proof has been given by
Zeilberger [37]. Dodgson (a.k.a. Lewis Carroll) devised a method called condensa
tion of determinants, which recursively calculates the determinant of a matrix using
3
only determinants of 2 x 2 matrices. This method was generalized to the notion of
A-determinants by Robbins and Rumsey. The study of A-determinants eventually led
Robbins and Rumsey to some groundbreaking work on alternating sign matrices and
resulted in Mills, Robbins, and Rumsey's famous ASM conjecture (see [7] and references
within).
In Chapter 3, we discuss Dodgson's Rule and how it follows from Muir's law of
extensible minors. By use of Muir's law and the determinant definition for a k x k
matrix, Dodgson's rule can be generalized to an identity we call the Dodgson/Muir
identity. We then extend Zeilberger's work to give a combinatorial proof of this identity.
Much like the determinant, the permanent of a matrix has one term for each of the
nonzero diagonals of the matrix, except each term counts positively. In 1913, Polya
[32] asked if it was possible to characterize the matrices A whose permanents are equal
to their determinants. This question carries great significance, as the calculation of
the permanent was shown by Valiant [36] to be a computationally difficult problem,
specifically what is called a #P-Complete problem. Calculating the determinant, on the
other hand, is a computationally simple problem and can be calculated in polynomial
time. Polya's problem leads naturally to sign-nonsingular (SNS) matrices. SNS matrices
are matrices which are determined to be nonsingular based solely upon the positions of
the zero, positive, and negative entries (see [11]). If A is SNS, then per(|A|) =±det(A),
and so we have a way of using a determinant to calculate a permanent.
Separately, Kasteleyn [20, 21, 22] worked with SNS matrices while studying the dimer
problem in statistical mechanics and conjectured one could find the number of perfect
matchings of a graph of genus g by using a linear combination of 49 Pfaffians; Galluccio
and Loebl [14] and Tesler [35] solved this. As a result, if A is the bipartite adjacency
4
matrix of a genus g bipartite graph, then 29 • per(A) can be written as a sum of the
determinants of A9 signings of A. Codenotti and Resta [12] use this result to calculate
the permanents for several families of sparse circulant matrices.
In Chapter 4, we will consider the structure of (0, l)-matrices that are maximal
with respect to having a SNS signing. Moreover, we investigate the calculation of the
permanent of a matrix and how to use determinants of several signings of the original
to accomplish our goal. Motivated by the work of Kasteleyn, we will then look at the
matrices J (n > 3), whose permanents can be computed using the determinants of
n
(fewer) matrices which are not necessarily signings of the original.
