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My research concerns aspects of enumerative, asymptotic, probabilistic and extremal combinatorics, particularly with relation to permutations.
See below for my preprints and publications, and slides from my talks.

Preprints

• Introducing irrational enumeration: analytic combinatorics for objects of irrational size
  (with Julien Condé)
• On cycles in monotone grid classes of permutations
  (with Robert Brignall and Nik Ruškuc)
  to appear in Combinatorial Theory

Publications

• On the asymptotic enumeration and limit shapes of monotone grid classes of permutations
  (with Noura Alshammari)
  Discrete Math. & Theor. Comput. Sci., 27(1), #2, 2025.
• On the evolution of random integer compositions
  (with Dan Threlfall)
  Electron. J. Combin., 32(1): P1.21, 2025.
• On naturally labelled posets and permutations avoiding 12–34
  (with Gi-Sang Cheon and Sergey Kitaev)
  European J. Combin., 126: Paper 104117, 2025.
• Thresholds for patterns in random permutations with a given number of inversions
  (with Dan Threlfall)
  Electron. J. Combin., 31(4): P4.6, 2024.
• Permutons and permutation class limit shapes
  Mini-Workshop: Permutation Patterns, 281–282, Mathematisches Forschungsinstitut Oberwolfach, 2024.
• 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.
• Permutation patterns: basic definitions and notation
  Unpublished, 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.
• FindPhone: Phonological analysis for the field linguist
  Summer Institute of Linguistics, 1995.
• What can you do with FindPhone?
  Notes on Linguistics 61:28–39, 1993.
• Output and efficiency of the closed-cycle gas turbine
  (with W. A. Woods and P. J. Bevan)
  Proc. Inst. Mech. Eng., Part A: Journal of Power and Energy, 205(1):59–66, 1991
• Principles for the design of a distributed memory architecture for parallel graph reduction
  (with G. L. Burn, R. J. Karia, and J. D. Robson)
  The Computer J., 32(5):461–469, 1989.
• An efficient reference counting solution to the distributed garbage collection problem
  Parallel Computing, 9(2):179–192, 1989.
• Overview of a parallel reduction machine project
  (with G. L. Burn, and R. J. Karia)
  PARLE Parallel Architectures and Languages Europe I, Lecture Notes in Comp. Sci. 258:394–413, 1987.
• Distributed garbage collection using reference counting
  PARLE Parallel Architectures and Languages Europe II, Lecture Notes in Comp. Sci. 259:176–187, 1987.
• Occam, semantics, and implementation
  Master’s thesis, University of Oxford, 1984.

Talks

• Monotone Grid Classes: Limit Shapes and Enumeration
  PP St Andrews Jul 2025
• Irrational enumeration: Analytic combinatorics for objects of irrational size
  Strathclyde Feb 2025 | Open University May 2025 | NORCOM Reykjavik Jun 2025 | ICECA Aug 2025
• Permutons and permutation class limit shapes
  Oberwolfach Jan 2024
• Mesh patterns in random permutations
  Dagstuhl Mar 2023 | PP Dijon Jul 2023 | Strathclyde Oct 2023
• Permutation limits at infinitely many scales
  PP Jun 2021 | BCC Lancaster Jul 2022
• What is a permuton and what is it good for?
  Strathclyde Mar 2021
• Limits of permutations (and some other discrete objects)
  Open University Dec 2020
• Permutations avoiding patterns and the curious behaviour of the total displacement
  Strathclyde Oct 2020
• The odd behaviour of the total displacement ratio
  PP Zurich Jun 2019
• The local structure of semi-sparse permutations
  PP Dartmouth Jul 2018 | SCM Edinburgh Apr 2019 | Strathclyde Mar 2022
• What can’t computers do?
  LectureFest Shrewsbury Jun 2018
• Staircases, dominoes and leaves: Bounds on gr(Av(1324))
  PP Reykjavik Jun 2017
• Prolific permutations and permuted packings
  SCM St Andrews Apr 2017 | BCC Strathclyde Jul 2017 | Open University Nov 2017
• Directionally convex lattice animals
  Strathclyde Jul 2016
• Geometric grid classes of permutations and the matching polynomial
  AMS Georgetown Mar 2015
• Expansions in non-integer bases and intervals of permutation class growth rates
  University of Florida Mar 2015 | Open University Mar 2015 | PP London Jun 2015
• The matching polynomial and geometric grid classes of permutations
  SEICCGTC Boca Raton Mar 2015
• The growth of permutations avoiding 1324
  Open University Feb 2015 | University of Florida Mar 2015 | BCC Warwick Jul 2015
• Two problems in discrete geometry: Some attractive applications of the probabilistic method
  Open University Dec 2014
• Tours on graphs and grid classes of permutations
  Warwick Oct 2014
• A fast-growing subset of Av(1324)
  PP East Tennessee Jul 2014
• Permutation grid classes and the spectra of graphs
  BCC Royal Holloway Jul 2013
• Enumerative combinatorics
  Open University Mar 2013
• Growth rates of permutation grid classes
  Open University Dec 2012 | Cambridge Nov 2013
• Skinny permutation grid classes
  PCC Warwick Aug 2012