Search

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher kby constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings.

There is a categorical equivalence between the Temperley--Lieb category $TL(2)$ and the full subcategory of $SU(2)$-\textbf{mod} with objects given by $V^{\otimes k}$ where $V$ is the tautological $SU(2)$-module and $k$ is a non-negative integer. The first results in this dissertation develop new diagrammatic categories which are shown to be equivalent to similarly defined full subcategories of $G$-\textbf{mod} for certain finite subgroups $G$ of $SU(2)$. The diagrams which generate the Temperley--Lieb category are shown to be linear combinations of the generating diagrams for these newly defined diagrammatic categories. The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit criteria, there is an equivalence of categories between a quotient category of $\mathbf{Dgrams}_{R(V,G)}$ and a full subcategory of $G-\textbf{mod}$ with objects being the tensor products of finitely many irreducible $G$-modules.

In 1974, Marcel Herzog and Jochanan Schönheim proposed “Research Problem no. 9” in The Canadian Mathematical Bulletin. The conjecture states that for a group G with non-trivial, finite left coset partition P = {g_iG_i}_{i=1}^k, the indices [G : G_1],[G : G_2],...,[G : G_k] cannot be distinct. This conjecture is referred to as the Herzog-Schönheim Conjecture. In this paper, we will discuss how the conjecture holds for various finite groups. We will also examine the work of Marc Berger, Alexander Felzenbaum, and Aviezri Fraenkel in exploring the conjecture’s relationship to finite pyramidal groups.

Given a simplicial complex $\Delta$ and its barycentric subdivision $\sdd$, we explore two homogeneous systems of parameters: one is the elementary symmetric functions inside the $\N$-graded Stanley-Reisner ring of $\Delta,$ $\k[\Delta],$ and the other is written based on a balanced coloring of $\sdd$ which lives inside the $\N^d$-graded Stanley-Reisner ring of $\sdd,$ namely, $\k[\sdd].$ The first is stable under symmetries and the other is stable under colorful automorphisms of $\sdd.$ In this paper, we develop methodology to explore comparing the resolutions of $\k[\Delta]$ and $\k[\sdd]$ over their respective parameter rings. We then prove it in the trivial case, which is simply the coinvariant algebra with alternate grading and one non-trivial case.

The following outlines my Ph.D thesis on Lagrange Inversion, Rogers-Ramanujan identities, and orthogonal polynomials. Sections $1-3$ serve as a summary of background material, much of which is based on various notes and papers either authored or co-authored by my advisor, Adriano Garsia. The subsequent sections contain related novel research. Specifically, section $4$ outlines the contents of a joint paper \cite{GG20} with Adriano Garsia, where we apply an expanded version of the combinatorial theory of orthogonal polynomials to produce new identities for the Fibonacci Polynomials. Section $5$ then details a purely combinatorial proof of $q-$Lagrange inversion, its application to Rogers-Ramanujan type identities, and more generalized Lagrange inversions involving an additional $t$ statistic.

In this thesis we provide an exposition of Szemere'dis theorem for the specific case of κ=4 which precedes Szemer\'{e}di's more famous result. The proof is elementary. The case of $k=3$, better known as Roth's theorem is proved as a corollary. Chapter 2 will introduce the lemmas required to prove Szemer\'{e}di's theorem for 4-terms, they are Lemmas~$BCDE$ and $(H_{0},\dots,H_{k})$. The chapter will conclude with a proof of the theorem. In chapter 3 we will prove Lemma $(H_{0},\dots,H_{k})$ along with 3 other necessary lemmas. These are the so called Simple Lemma, Lemma $p(\delta,\ell)$, and Lemma $|G^*|$ respectively. Afterwards we discuss how, with some modification to Lemma $|G^*|$, we may formulate and prove Roth's theorem as a corollary. Finally, chapter 4 will be entirely on the proof of Lemma~$BCDE$. Being the most involved in Szemeredi's construction, it will use several arguments made throughout the paper in addition to new ideas.

In 1980, Murray Marshall proved that the category of Quaternionic Structures is naturally equivalent to the category of abstract Witt rings. This paper develops a combinatorial theoryfor finite Quaternionic Structures in the case where $1=-1$,by demonstrating an equivalence between finite quaternionic structures and Steiner Triple Systems (STSs) with suitable blockcolorings. Associated to these STSs are Block Intersection Graphs (BIGs)with induced vertex colorings. This equivalence allows for a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associatedto the Witt rings of so-called elementary type, by providingnecessary and sufficient conditions for a BIG associated toa product or group extension.

The Egyptian Fractions of One problem (EFO), asks the following question: Given a positive integer n, how many ways can 1 be expressed as the sum of n non-increasing unit fractions? In this paper, we verify a result concerning the EFO problem for n=8, and show the computational complexity of the problem can be severely lessened by new theorems concerning the structure of solutions to the EFO problem.

We examine the Carlson-Simpson Lemma (VW(k,l)), which is the combinatorial core of the Dual Ramsey Theorem, from the perspective of Reverse Mathematics. Our results include the following:Working in the system BSigma_2^0, we carry out the construction of a failure of the ordered version of the Carlson-Simpson Lemma OVW(k,l), which was introduced in a paper by Miller and Solomon. This observation implies that we can construct such a recursive counterexample in the model of SRT_2^2 that was discussed in a recent paper by Chong, Slaman and Yang. It follows that SRT_2^2 does not prove OVW(k,l) over RCA_0.We also show that the strength of the principle VW(k,l) is independent of the number of colors l being used.By proving that VW(k,l) is not conservative over RCA_0 for arithmetical sentences, we conclude that VW(k,l) is not provable from any theory that is conservative over RCA_0 for arithmetical sentences.

Our goal is to be able to determine the distance between two partitions in the same Rouquier block. This paper explores a lower bound on the number of steps required to get from the quotient of a partition lambda; to (empty set, empty set, ..., (w)). We will use the Young diagrams in the quotient of lambda and a process of right and left steps to obtain a new partition that is connected to lambda in the Brauer graph of these representations. Paired with this process we will incorporate a notation called the badness of each box in the Young diagram representation of the quotient of lambda;. This will allow us to show that the lower bound on the number of steps is less than or equal to the initial badness, b, of the starting partition divided by the maximum reduction of badness by each right and left step, 2w^2.

In this paper, we examine a certain graph related to the representation theory of the symmetric group. This graph is called the Brauer graph of a Rouquier block. It is known that the diameter d(R) of this graph satisfies the following condition: the diameter d(R) is between p - 1 and two times the ceiling function of "log base two of w" + p + 2 where p and w are certain associated parameters. Note that the preceding condition describes both a lower bound and an upper bound on the diameter d(R). We conjecture that the best upper bound for the diameter is actually between p - 1 and the ceiling function of "log base 2 of w" + p. We conjecture that two vertices in our graph have the maximum distance between them, and we prove that this distance is at most the ceiling function of "log base 2 of w" + p by exhibiting an algorithm that creates a path between these two vertices.

This paper solves a problem faced by the Suburban East Conference of the Minnesota State High School League in 2009, of designing a consistent wrestling schedule to accommodate a new school. We adapt the application of projective geometry of finite fields to general scheduling problems, and develop an algorithm for determining a schedule. We prove that this algorithm can be completed, will yield the desired schedule, and can yield all possible schedules in the desired format. We also model the schedule with bipartite graphs, and use edge colorings to complete the schedule with home and away assignments.

This paper develops a data structure based on preimage sets of functions on a finite set. This structure, called the sigma matrix, is shown to be particularly well-suited for exploring the structural characteristics of recursive functions relevant to investigations of complexity. The matrix is easy to compute by hand, defined for any finite function, reflects intrinsic properties of its generating function, and the map taking functions to sigma matrices admits a simple polynomial-time algorithm . Finally, we develop a flexible measure of preimage complexity using the aforementioned matrix. This measure naturally partitions all functions on a finite set by characteristics inherent in each function's preimage structure.

Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with a natural involution. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's ``$q=-1$ phenomenon''. A polynomial $\sum_{i=0}^{n} a_i t^i$ with symmetric coefficients ($a_{n-i}=a_i$) has a unique expansion $\sum_{k=0}^{\lfloor n/2 \rfloor} \gamma_k t^k(1+t)^{n-2k}$, and is said to be \emph{gamma-nonnegative} if $\gamma_k\geq 0$ for all $k$. We either prove or conjecture a stronger $q$-analogue of this property for several polynomials in two variables $t$,$q$, whose $q=1$ specializations are known to be gamma-nonnegative.

Inspired by an investigation of Ramsey theory, this paper aims to clarify in further detail a number of results regarding the existence of monochromatic cycles in complete graphs whose edges are colored red or blue. The second half focuses on a proof given by Gyula Károlyi and Vera Rosta for the solution of all R(Cn,Ck).

In this paper we look at two (unrelated) problems. In the first problem we study a class of symmetric matrices $T$ indexed by positive integers $m\ \geq \ n \geq \ 2$, with the objective of finding the eigenvalues and eigenspaces of $T$. The nonsingularity of $T$ implies Foulkes's Conjecture (for these values of $m$ and $n$.) In the case $n$ = 2 we completely determine the eigenvalues and eigenspaces of $T$ and in so doing demonstrate the nonsingularity of $T$. For $n$ = 3 we develop a fast algorithm for computing the eigenvalues of $T$, and give numerical results in the cases $m$ = 3, 4, 5. In the second problem we find the eigenvalues and eigenvectors of the Laplacian of the chain complex ($C\sb\*(P),\partial\sb\*$) of a finite poset $P$ which we describe as follows: the elements of $P$ are arranged in $p$ levels $L\sb1$,$L\sb2,\...,L\sb{p},$ from bottom to top, and for any integer $i$ with 2 $\leq\ i\ \leq \ p$, $x$ $