Discrete Mathematics and Combinatorics Commons^™

Circulant Matrices And Mathematical Juggling, Richard A. Brualdi, Michael W. Schroeder

Mathematics Faculty Research

Circulants form a well-studied and important class of matrices, and they arise in many algebraic and combinatorial contexts, in particular as multiplication tables of cyclic groups and as special classes of latin squares. There is also a known connection between circulants and mathematical juggling. The purpose of this note is to expound on this connection developing further some of its properties. We also formulate some problems and conjectures with some computational data supporting them.

Covering Arrays For Equivalence Classes Of Words, Joshua Cassels, Anant Godbole

Undergraduate Honors Theses

Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the d^t t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds …

Educational Magic Tricks Based On Error-Detection Schemes, Ronald I. Greenberg

Ronald Greenberg

Magic tricks based on computer science concepts help grab student attention and can motivate them to delve more deeply. Error detection ideas long used by computer scientists provide a rich basis for working magic; probably the most well known trick of this type is one included in the CS Unplugged activities. This paper shows that much more powerful variations of the trick can be performed, some in an unplugged environment and some with computer assistance. Some of the tricks also show off additional concepts in computer science and discrete mathematics.

Educational Magic Tricks Based On Error-Detection Schemes, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

Magic tricks based on computer science concepts help grab student attention and can motivate them to delve more deeply. Error detection ideas long used by computer scientists provide a rich basis for working magic; probably the most well known trick of this type is one included in the CS Unplugged activities. This paper shows that much more powerful variations of the trick can be performed, some in an unplugged environment and some with computer assistance. Some of the tricks also show off additional concepts in computer science and discrete mathematics.

Distribution Of Permutation Statistics Across Pattern Avoidance Classes, And The Search For A Denert-Associated Condition Equivalent To Pattern Avoidance, Joshua Thomas Agustin Davies

Dissertations, Master's Theses and Master's Reports

We begin with a discussion of the symmetricity of $\maj$ over $\des$ in pattern avoidance classes, and its relationship to $\maj$-Wilf equivalence. From this, we explore the distribution of permutation statistics across pattern avoidance for patterns of length 3 and 4.

We then begin discussion of Han's bijection, a bijection on permutations which sends the major index to Denert's statistic and the descent number to the (strong) excedance number. We show the existence of several infinite families of fixed points for Han's bijection.

Finally, we discuss the image of pattern avoidance classes under Han's bijection, for the purpose of finding …

Pattern Containment In Circular Permutations, Charles Lanning

Electronic Theses and Dissertations

Pattern containment in permutations, as opposed to pattern avoidance, involves two aspects. The first is to contain every pattern at least once from a given set, known as finding superpatterns; while the second is to contain some given pattern as many times as possible, known as pattern packing. In this thesis, we explore these two questions in circular permutations and present some interesting observations. We also raise some questions and propose some directions for future study.

Combinatorial Potpourri: Permutations, Products, Posets, And Pfaffians, Norman B. Fox

Theses and Dissertations--Mathematics

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their cd-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two cd-monomials. Then the cd-index is defined for infinite posets, with …

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

Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi

HMC Senior Theses

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θ^K, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The m^th moment of the r^th longest cycle of a random permutation is Θ(n^m), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or …

The Link Between Scrambling Numbers And Derangements, Barry Balof, Eric Farmer, Jamie Kawabata

Mathematical Sciences Technical Reports (MSTR)

The group equation abcdef = dabecf can be reduced to the equation xcde = dxec. In general, we are interested in how many variables are needed to represent group equations in which the right side is a permutation of the variables on the left side. Scrambling numbers capture this information about a permutation. In this paper we present several facts about scrambling numbers, and expose a striking relationship between permutations that cannot be reduced and derangements.