Discrete Mathematics and Combinatorics Commons^™

Adjoint Appell-Euler And First Kind Appell-Bernoulli Polynomials, Pierpaolo Natalini, Paolo E. Ricci

Applications and Applied Mathematics: An International Journal (AAM)

The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed.

Cycle Structures Of Orthomorphisms Extending Partial Orthomorphisms Of Boolean Groups, Nichole Louise Schimanski, John S. Caughman Iv

Mathematics and Statistics Faculty Publications and Presentations

A partial orthomorphism of a group GG (with additive notation) is an injection π:S→G for some S⊆G such that π(x)−x ≠ π(y) for all distinct x,y∈S. We refer to |S| as the size of π, and if S=G, then π is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have.

It is known that conjugation by automorphisms of G forms a group action on the set of orthomorphisms of G. In this paper, we consider the …

Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

Student Research Symposium

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.

A Study Of Graphical Permutations, Jessica Thune

UNLV Theses, Dissertations, Professional Papers, and Capstones

A permutation π on a set of positive integers {a_1,a_2,...,a_n} is said to be graphical if there exists a graph containing exactly a_i vertices of degree (a_i) for each i. It has been shown that for positive integers with a_1

Generalizations Of Pascal's Triangle: A Construction Based Approach, Michael Anton Kuhlmann

UNLV Theses, Dissertations, Professional Papers, and Capstones

The study of this paper is based on current generalizations of Pascal's Triangle, both the expansion of the polynomial of one variable and the multivariate case. Our goal is to establish relationships between these generalizations, and to use the properties of the generalizations to create a new type of generalization for the multivariate case that can be represented in the third dimension.

In the first part of this paper we look at Pascal's original Triangle with properties and classical applications. We then look at contemporary extensions of the triangle to coefficient arrays for polynomials of two forms. The first of …

Zero-Sum Magic Graphs And Their Null Sets, Samuel M. Hansen

UNLV Theses, Dissertations, Professional Papers, and Capstones

For any element h of the Natural numbers, a graph G=(V,E), with vertex set V and edge set E, is said to be h-magic if there exists a labeling of the edge set E, using the integer group mod h such that the induced vertex labeling, the sum of all edges incident to a vertex, is a constant map. When this constant is 0 we call G a zero-sum h-magic graph. The null set of G is the set of all natural numbers h for which G admits a zero-sum h-magic labeling. A graph G is said to be uniformly …

Snort: A Combinatorial Game, Keiko Kakihara

Theses Digitization Project

This paper focuses on the game Snort, which is a combinatorial game on graphs. This paper will explore the characteristics of opposability through examples. More fully, we obtain some neccessary conditions for a graph to be opposable. Since an opposable graph guarantees a second player win, we examine graphs that result in a first player win.

Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger

CMC Senior Theses

Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.

Nonlinear Dynamics In Combinatorial Games: Renormalizing Chomp, Eric J. Friedman, Adam S. Landsberg

WM Keck Science Faculty Papers

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to …

Freeness Of Hopf Algebras, Christopher David Walker

Theses Digitization Project

The Nichols-Zoeller freeness theorem states that a finite dimensional Hopf algebra is free as a module over any subHopfalgebra. We will prove this theorem, as well as the first significant generalization of this theorem, which was proven three years later. This generalization says that if the Hopf algebra is infinite dimensional, then the Hopf algebra is still free if the subHopfalgebra is finite dimensional and semisimple . We will also look at several other significant generalizations that have since been proven.

The Terwilliger Algebra Of An Almost-Bipartite P- And Q-Polynomial Association Scheme, John S. Caughman Iv, Mark S. Maclean, Paul M. Terwilliger

Mathematics and Statistics Faculty Publications and Presentations

Let Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the …

The Multiplicities Of A Dual-Thin Q-Polynomial Association Scheme, Bruce E. Sagen, John S. Caughman Iv

Mathematics and Statistics Faculty Publications and Presentations

Let Y=(X,{Ri}1≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E₀,...,E_D of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi (0≤i≤D) of Yis unimodal. Talking to Terwilliger, Stanton made the related conjecture that mi≤mi+1 and mi≤mD−i for i<D/2. We prove that if Y is dual-thin in the sense of Terwilliger, then the …