# David Bevan

Lecturer B

### Personal Statement

I am a Lecturer in Combinatorics in the Department of Computer and Information Sciences.

My research interests are in enumerative and extremal combinatorics, particularly in relation to permutations.

Recent research has resulted in the first general result concerning the exact growth rates of a broad family of permutation classes, published in the *Transactions of the American Mathematical Society. *Another paper, published in the *Journal of the London Mathematical Society*, uses a novel approach to yield an improved lower bound to the notorious problem of computing the growth rate of the class of permutations avoiding the pattern 1324.

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“Enumeration, otherwise known as counting, is the oldest mathematical subject.”

— Doron Zeilberger, *The Princeton Companion to Mathematics*, 2008

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“A generating function is a clothesline on which we hang up a sequence of numbers for display.”

— Herbert Wilf, *generatingfunctionology*, 1994

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In the 1980s, following undergraduate studies in mathematics at the University of Oxford, I undertook some computer science research. For my Oxford M.Sc. dissertation, I developed a model for the denotational semantics of the concurrent programming language occam. Following this, I spent two years in industry with GEC, working on formalisms for specifying communications protocols, and, as part of a project to design a parallel machine to run functional programming languages, produced a paper that introduced weighted reference counting, now a key method for managing memory in distributed computer architectures.

This was followed by a career in software development, first as a developer, consultant and trainer for the Summer Institute of Linguistics, based in Papua New Guinea, and subsequently as a software engineer and development manager with Emtex Ltd. (acquired by Pitney Bowes in 2006).

A decade ago, I carried out some independent mathematical research in my spare time, resulting in the publication of a paper improving on a long-standing extremal result of Erdős and Füredi in discrete geometry.

In 2012, I left software development for full-time mathematical research, and in June 2015 was awarded a PhD in enumerative combinatorics from The Open University. My supervisor was Robert Brignall. The topic of my thesis is the growth of permutation classes. As a result of my doctoral studies, I produced five papers.

After completing my PhD, I was a Visiting Research Fellow at The Open University for a year, during which time I completed four more papers. Two of these concern the degree-diameter problem for Cayley graphs. The other two address specific enumerative questions in pattern-avoidance.

### Research Interests

My research interests concern aspects of enumerative, asymptotic and extremal combinatorics, particularly with relation to permutations.

#### Preprints

- Prolific permutations and permuted packings: downsets containing many large patterns (with Cheyne Homberger and Bridget Tenner)
- Large "De Bruijn" Cayley graphs and digraphs, to appear in
*Discrete Mathematics*. - The permutation class Av(4213,2143), to appear in
*Discrete Math. & Theor. Comput. Sci*.

#### Recent publications

- Intervals of permutation class growth rates,
*Combinatorica*, 2017. - Large circulant graphs of fixed diameter and arbitrary degree (with Grahame Erskine and Rob Lewis),
*Ars Mathematica Contemporanea*, 13(2):275–291, 2017. - Pattern avoidance in forests of binary shrubs (with Derek Levin, Peter Nugent, Jay Pantone, Lara Pudwell, Manda Riehl and ML Tlachac),
*Discrete Math. & Theor. Comput. Sci*., 18(2), #8, 2016. - The permutation classes Av(1234,2341) and Av(1243,2314),
*Australasian J. Combin.*, 64(1):3–20, 2016. - Permutations avoiding 1324 and patterns in Łukasiewicz paths,
*J. London Math. Soc.*, 92(1):105–122, 2015. - Growth rates of permutation grid classes, tours on graphs, and the spectral radius,
*Trans. Amer. Math. Soc.*, 367(8):5863–5889, 2015. - Growth rates of geometric grid classes of permutations,
*Electron. J. Combin.*, 21(4): P4.51, 2014. - Sets of points determining only acute angles and some related colouring problems,
*Electron. J. Combin.*, 13(1): R12, 2006.

Details of all my publications can be found on my Google Scholar page.

#### Slides from talks

- Expansions in non-integer bases and intervals of permutation class growth rates, 9 March 2015, University of Florida.
- The matching polynomial and geometric grid classes of permutations, 3 March 2015 at 46th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University.
- Two problems in discrete geometry, 17 December 2014, The Open University.
- A fast-growing subset of Av(1324), 7 July 2014 at Permutation Patterns 2014, East Tennessee State University.
- Permutation grid classes and the spectra of graphs, 4 July 2013 at 24th British Combinatorial Conference, Royal Holloway University of London.
- Enumerative combinatorics, 14 March 2013, The Open University.
- Skinny permutation grid classes, 17 August 2012 at 22nd Postgraduate Combinatorial Conference, University of Warwick.

#### Mathematica demonstrations

## Publications

(10)## Large circulant graphs of fixed diameter and arbitrary degree

Research output: Contribution to journal › Article

## The permutation class Av(4213,2143)

Research output: Contribution to journal › Article

## Pattern avoidance in forests of binary shrubs

Research output: Contribution to journal › Article

## Activities

(1)## Permutation Patterns 2015

Activity: Participating in or organising an event › Organiser of major conference