I am a Lecturer in Combinatorics. My research interests concern enumerative, asymptotic and extremal questions, particularly in relation to permutations.

Enumerative combinatorics is concerned with counting, either exactly or approximately, the number of discrete structures satisfying certain constraints. Asymptotic combinatorics is to do with determining the structure and properties of typical large discrete objects. Extremal combinatorics concerns determining the size of the largest possible discrete structures of a given type.

Current topics of research include enumerative and structural questions concerning grid classes of permutations, the enumeration and structure of the class of permutations avoiding the pattern 1324, and how the structure of a random permutation evolves as the number of its inversions increases.

For more information about research in combinatorics at the University of Strathclyde, see the Strathclyde Combinatorics Group webpage.

A bit of background

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, during which I 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 in the voluntary sector, based in Papua New Guinea, and subsequently as a software engineer and development manager in industry in the UK.

In my spare time, I carried out some independent mathematical research 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 2015 was awarded a PhD from The Open University. The topic of my thesis was the growth of permutation classes. Following a year as a Visiting Research Fellow and Associate Lecturer at The Open University, I took up my current position in September 2016.

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

Preprints

• On naturally labelled posets and permutations avoiding 12–34 (with Gi-Sang Cheon and Sergey Kitaev).
• On the evolution of random integer compositions (with Dan Threlfall).

Recent publications

• Threshold functions and the birthday paradox, Math. Gaz. 106(566):343–348, 2022.
• Independence of permutation limits at infinitely many scales, J. Combin. Theory A. 186: Paper 105557, 2022.
• A structural characterisation of Av(1324) and new bounds on its growth rate (with Robert Brignall, Andrew Elvey Price and Jay Pantone), European J. Combin. 88: Paper 103115, 2020.
• Bijections between directed animals, multisets and Grand-Dyck paths (with Jean-Luc Baril and Sergey Kirgizov). Electron. J. Combin., 27(2): P2.10, 2020.
  
• Intervals of permutation class growth rates, Combinatorica, 38(2):279–303, 2018.
• Prolific permutations and permuted packings: downsets containing many large patterns (with Cheyne Homberger and Bridget Tenner), J. Combin. Theory A., 153:98–121, 2018.
• Staircases, dominoes, and the growth rate of 1324-avoiders (with Robert Brignall, Andrew Elvey Price and Jay Pantone), Electron. Notes in Discrete Math., 61:123–129, 2017.
• Large butterfly Cayley graphs and digraphs, Discrete Math., 340(10):2432–2436, 2017.
• The permutation class Av(4213,2143), Discrete Math. & Theor. Comput. Sci., 18(2): #10, 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.
• On the growth of permutation classes, PhD thesis, The Open University, 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.

My older publications can be found on the following pages: ORCiD / dblp / Scopus / Google Scholar

Slides from talks

• Limits of permutations (and some other discrete objects), 16 December 2020, The Open University Discrete Mathematics Seminar.
• The curious behaviour of the total displacement, 6 November 2018, at Genomics, Pattern Avoidance, and Statistical Mechanics, Schloss Dagstuhl.
• The local structure of semi-sparse permutations, 9 July 2018, at Permutation Patterns 2018, Dartmouth College.
• Staircases, dominoes and leaves: Bounds on gr(Av(1324)), 29 June 2017, at Permutation Patterns 2017, Reykjavík University.
• Prolific permutations and permuted packings, 25 April 2017, at Scottish Combinatorics Meeting, University of St Andrews.
• 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

• Permutation Grid Classes
• Open Meanders
• Random Closed Meanders

Currently, I'm teaching the following:

MM109 Applying Mathematics 2: Graph Theory

Types of graphs; graph operations, walks on graphs; connectivity; Eulerian graphs; Hamiltonian graphs; algorithms for weighted graphs; trees and forests; spanning trees; planarity; colouring; matchings; digraphs; network flows.

MM917 Networks in Finance

Random networks; the small-world phenomenon; scale-free networks; the Barabási–Albert model; centrality; degree correlation; robustness; spreading; communities.

Past teaching responsibilities include the following:

MM116 Mathematics 1c (Semester 1)

Foundations; introduction to calculus.

CS103 Machines, Languages and Computation (Semester 2)

Propositional logic and proofs using natural deduction; normal forms and satisfiability; computational complexity, P and NP; finite state automata and regular expressions, the Brzozowski algebraic method and the pumping lemma; Turing machines, undecidability, the halting problem and the Entscheidungsproblem.

CS104 Information and Information Systems (Module 1: Information Theory)

Data and information (syntax and semantics, text encodings, Unicode and UTF-8); error detection and correction (repetition codes, parity bits and Hamming codes); data compression (run-length encoding and LZW); measuring information (entropy) and Shannon's Source Coding Theorem.

CS106 Computer Systems and Organisation (Semester 2)
CS107 Fundamentals of Computer Systems

Computer organization; MIPS assembly programming and the MIPS Instruction Set Architecture (registers, memory addressing, logical and shifting operations, jumps and branches, loops and arrays, integers and integer arithmetic, subroutines and the call stack, recursion); memory caching; virtual memory.