#### Title

Gdd(N1+N2, 3; Λ1, Λ2) With Equal Number Of Blocks Of Two Configurations

#### School Name

Governor's School for Science and Math

#### Grade Level

12th Grade

#### Presentation Topic

Math and Computer Science

#### Presentation Type

Mentored

#### Abstract

A GDD(n1+n2, 3; λ1, λ2) is a group divisible design with two groups of sizes n1 and n2 with block size 3 such that each pair of distinct elements from the same group occurs in λ1 blocks and each pair of elements from different groups occurs in λ2 blocks. We prove that necessary conditions are sufficient for the existence of group divisible designs GDD(n1+n2, 3; λ1, λ2) with equal number of blocks of configuration (1,2) and (0,3) for n1+n2≤20 and in general for n1=1,2,4, n2-1 and n2-2. We also give near complete results of n1=3.

#### Recommended Citation

Billings, Blaine, "Gdd(N1+N2, 3; Λ1, Λ2) With Equal Number Of Blocks Of Two Configurations" (2016). *South Carolina Junior Academy of Science*. 83.

https://scholarexchange.furman.edu/scjas/2016/all/83

#### Location

Owens 207

#### Start Date

4-16-2016 8:45 AM

Gdd(N1+N2, 3; Λ1, Λ2) With Equal Number Of Blocks Of Two Configurations

Owens 207

A GDD(n1+n2, 3; λ1, λ2) is a group divisible design with two groups of sizes n1 and n2 with block size 3 such that each pair of distinct elements from the same group occurs in λ1 blocks and each pair of elements from different groups occurs in λ2 blocks. We prove that necessary conditions are sufficient for the existence of group divisible designs GDD(n1+n2, 3; λ1, λ2) with equal number of blocks of configuration (1,2) and (0,3) for n1+n2≤20 and in general for n1=1,2,4, n2-1 and n2-2. We also give near complete results of n1=3.

## Mentor

Mentor: Dr. Sarvate; Department of Mathematics, College of Charleston