Combinatorics Abstracts

On Two-Circulant Weighing Matrices

K.T. Arasu, Wright State University

We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W(148; 144), W(152;144), W(156; 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. In addition, we obtain infinite families of weighing matrices constructed from two circulants, such as W(68 + 2k; 52) and W(108 + 2k; 64) for all k > 0, based on ternary complementary pairs. We also fill a missing entry in Strassler's table with answer "YES", by constructing a circulant weighing matrix of order 142 with weight 100. This is joint work with Ilias S. Kotsireas, Christos Koukouvinos and Jennifer Seberry.

DAYTON, OH
k.arasu at wright.edu

Search for Good Linear Codes

Nuh Aydin, Kenyon College

This talk gives an overview of some of the recent search methods that have been successful in finding new linear codes. It also presents a recent search algorithm and the new codes obtained by that algorithm.

319 HAYES HALL
201 N COLLEGE RD
GAMBIER, OH 43022
aydinn at kenyon.edu

A $\lambda$ -Large Type Theorem for Candelabra Systems

Niranjan Balachandran, The Ohio State University

Candelabra systems are combinatorial objects that are closely related to Steiner designs. In this talk, we look at an appropriate incidence matrix that would define a Candelabra system as integers solutions to a certain matrix equation. We prove that the matrix in consideration has full rank. This has interesting consequences regarding some constructions for Steiner $3$ -designs.

DEPARTMENT OF MATHEMATICS
231 W 18TH AVE
COLUMBUS, OH 43210
niranj at math.ohio-state.edu

Cranks - Really the Final Problem

Bruce Carl Berndt, University of Illinois

The existence of the crank, a partition statistic to explain Ramanujan's famous congruence for the partition function modulo 11, was conjectured by Freeman Dyson in 1944. It was discovered by George Andrews and Frank Garvan on June 6, 1988. It was then noticed that the crank can be found in Ramanujan's lost notebook. We have not yet learned the meaning of various entries in the lost notebook pertaining to cranks. A survey of Ramanujan's work on cranks in his lost notebook will be given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks -- not mock theta functions.

1409 WEST GREEN ST.
URBANA, IL 61801
berndt at math.uiuc.edu

Classification of Orthogonal Arrays by Integer Programming

Dursun Bulutoglu, Air Force Institute of Technology

Factorial designs are used extensively in a wide range of scientific and industrial investigations for extracting as much information as possible at a fixed cost. Statisticians are interested in finding orthogonal arrays, since orthogonal arrays are the most efficient factorial designs for certain statistical models. However, finding an orthogonal array that is universally optimal for a statistical model is a difficult, unsolved problem. In fact, mathematicians have worked on variants of this problem since the time of Euler. Since the statistical properties of orthogonal arrays are preserved under design isomorphism, classifying them up to isomorphism allows the best to be found with respect to the statistical criterion of choice. A new method for finding all non-isomorphic factorial designs in a given set will be compared to those in the literature.

In Bulutoglu and Margot (2007), the problem of classifying all isomorphism classes of orthogonal arrays is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of those equations. Cases were solved using Margot's (2007) extension of the branch-and-cut algorithm. A new method for solving the same problem based on solving a sequence of ILPs using Margot's (2007) algorithm will be introduced. Pros and cons of the new method will be discussed. I will also provide a summary of other research directions for finding a best factorial design under the well established GMA statistical criterion.

DAYTON, OH
dursun.bulutoglu at gmail.com

Relative difference sets from additive Hadamard cocycles

Yuqing Chen, Wright State University

Additive Hadamard cocycle are a natrual generalization of presemifield in finite geomtry. In this talk, we will discuss relative difference sets obtained from additive Hadamard cocycles, including falg transitivity and absolute polarity of the divisible designs developed from these difference sets.

DAYTON, OH 45435
yuqing.chen at wright.edu

The Kuratowski covering conjecture for graphs of order $<$ 10

Suhkjin Hur, The Ohio State University

Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either $K_5$ or $K_{3,3}$ , called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite graph embeds in the nonorientable surface of genus $\tilde{g}$ if it does not contain $\tilde{g}+1$ Kuratowski subgraphs such that the union of each pair of these fails to embed in the projective plane, the union of each triple of these fails to embed in the Klein bottle if $\tilde{g} \ge 2$ , and the union of each triple of these fails to embed in the torus if $\tilde{g} \ge 3$ . We prove this conjecture for all graphs of order $<$ 10.

DEPARTMENT OF MATHEMATICS
231 W 18TH AVE
COLUMBUS, OH 43210
sjin at math.ohio-state.edu

Quadratic q-series

Verne E. Leininger, Bridgewater College

$q$ -factorials are usually indexed by a linear function. This presetation will discuss work in progress to use a quadratic function to index $q$ -factorials and the goal of finding the number of ways to express an integer as the sum of two cubes.

402 E COLLEGE STREET BRIDGEWATER, VA 22812
vleining at bridgewater.edu

An equivalence of Ward's bound and

its application to nonlinear divisible codes

Xiaoyu Liu, Wright State University

We prove an equivalent condition of Ward's bound on dimension of divisible codes, and conclude that Ward's bound is a consequence of the fact that the MacWilliams transform of the weight enumerator has integer coefficients. This equivalence generalizes Ward's bound to some nonlinear codes as long as the MacWilliams identities hold.

DAYTON, OH
xliu at noether.math.wright.edu

The Clique Chromatic Index of Line Graphs

Christopher Matthew McClain, The Ohio State University

The chromatic index of a graph $G$ is most often defined to be the minimum size of a partition of the edge set of $G$ into matchings. An equivalent definition is the minimum size of a cover of the edge set of $G$ by matchings. We consider the analogous problem of covering the edge set of a simple graph $G$ by subgraphs that are vertex-disjoint unions of cliques. We denote by $\chi_E^K(G)$ the minimum size of such a covering set, and investigate the special case $\chi_E^K(L(G))$ , where $L(G)$ is the line graph of $G$ .

DEPARTMENT OF MATHEMATICS
231 W 18TH AVE
COLUMBUS, OH 43210
mcclain at math.ohio-state.edu

Graph games with restricted degrees

Nishali Mehta, The Ohio State University

A. Hajnal proposed the following graph game: Starting with an empty graph, two players alternately draw edges. They are not allowed to complete a triangle and whoever cannot move is the loser. We study a version of the game with the added restriction that the graph created by the players has prescribed maximal valency.

DEPARTMENT OF MATHEMATICS
231 W 18TH AVE
COLUMBUS, OH 43210
nishali at math.osu.edu

Sums of squares and the Leech lattice

Stephen Carl Milne, The Ohio State University

Utilizing classical elliptic function invariants, we first sketch our derivation of several useful new formulas for Ramanujan's $\tau$ function. This work includes: the main pair of new formulas for the $\tau$ function that ``separate'' the two terms in the classical formula for the modular discriminant, a generating function form for both of these formulas, a Leech lattice form of one of these formulas, and a triangular numbers form. We then present analogous new formulas for several other classical cusp forms that appear in quadratic forms, sphere-packings, lattices and groups. An additional application to the theory of quadratic forms is also given.

DEPARTMENT OF MATHEMATICS
231 W. 18TH AVE
COLUMBUS, OH 43210
milne at math.ohio-state.edu

Juggling sequences with restricted heights

Jon Derek Stadler, Capital University

A juggling function is a bijection $f$ on the integers with the property that $f(t)\geq t$ for all $t$ . To each periodic juggling function, we can associate a finite sequence of integers, called heights. We will discuss a general method for enumerating juggling sequences in which restrictions are placed on these heights, thus addressing an open question recently posed by Graham and Chung. In honor of Steve Milne's birthday, there will be a demonstration of juggling 60.

COLUMBUS, OH
jstadler at capital.edu

On $k$ -splitting families

Xiangqian Zhou, Wright State University

A $k$ -splitting family consists of a set $E$ of size $2k+1$ and a collection $\mathcal{F}$ of $k$ -subsets of $E$ such that (1) two distinct subsets in $\mathcal{F}$ meet by at most $k-2$ elements; and for every $e \in E$ , there exists a partition $(A,B)$ of $E\backslash e$ with $A , B \in \mathcal{F}$ . We came up with this object when we worked on a matroid theory problem. In this talk, I will show some interesting relationship between $k$ -splitting families and $k$ -connected matroids. This is joint work with James Reid and Haidong Wu.

DAYTON, OH 45435
x.zhou at marshall.edu