## Thomas Robinson

#### Biography

I am currently enrolled at the University of Newcastle in B Maths/B Science.  I have completed two years of full time study.

Pure Mathematics is my major academic interest. In Science I am taking all Physics subjects.

My research interest over the summer vacation has been in Combinatorics.  I have been studying the 1324 avoiding permutations.

Non-academic interests include chess and computer games.

#### 1324 Avoiding Permutations

The goal of this research project is to study the 1324-avoiding permutations with a view to more tightly finding their bound limits (L). It is envisaged that this might be achieved by encoding the 1324-avoiding permutations in terms of a formal language.We will be taking as a starting point the recent paper written by Miklos Bona which lowered the known upper bound of L, by using a refined counting argument for the 1324 avoiding permutations in an interesting way. In this paper Bona first colours the 1324 avoiding permutations with a red-blue colouring, and then assigns a letter (A, B, C or D) to each entry of the permutation according to a defined set of rules. He then established an injection between this space and another space, allowing an upper bound of L to be established. He then refined the counting argument by introducing an additional rule, which allowed the range space of the injection to be reduced, leading to a lowering of the known upper bound of L. There is a lower bound of L found in 2005. This was recently been improved in a paper written by David Bevan. The bounds are still quite far apart, so even a small improvement is progress.The research method proposed is to study Bona’s proof carefully, under the guidance of my supervisor, Dr Murray Elder, and to find and study additional references in order to develop ideas, concepts and contexts useful for further progressing this work. This will involve extending and incorporating the familiar concepts presented in undergraduate discrete mathematics courses to this more specific setting. At the conclusion of the project our techniques and results will be presented to fellow mathematicians at the AMSI Big Day In.