Robert Brignall

Senior Lecturer in Combinatorics


I am a Senior Lecturer in Combinatorics in the School of Mathematics and Statistics at The Open University. My research interests primarily focus on the study of infinite antichains, well-quasi-ordering, permutation patterns, and the interplay between permutation patterns and similar concepts in graph theory. You may read more about my research, view my list of publications or my list of talks.

I am the school's Research Director, and I also organise our Seminar Series.

Previously, I was a research fellow in the Department of Mathematics at the University of Bristol, and before that I was a PhD student in the School of Mathematics and Statistics at the University of St Andrews.

In the past, I organised the Bristol Combinatorics Seminar, I was on the organising committees for Techniques and Problems in Graph Theory, Permutation Patterns 2010 and Permutation Patterns 2011, and I chaired the local organising committee for Permutation Patterns 2015.


My research interests broadly lie in the study of combinatorial and relational structures, in a blend of structural, extremal and enumerative combinatorics. My background is in the study of permutation containment and avoidance, where enumeration is the main game. However, the structural study of permutations can often be translated to other combinatorial objects, most notably graphs, where the consequences can be wide-ranging. Currently, I am especially investigating the question of well-quasi-ordering in combinatorial objects and the corresponding construction of infinite antichains, but I also have ongoing projects in permutation enumeration, and structural graph theory.

I submitted my PhD Thesis, entitled "Simplicity in Relational Structures and its Application to Permutation Classes", in July 2007, and successfully defended it in October 2007. My external examiner was Einar Steingrímsson (Strathclyde University), my internal Steve Linton. My PhD was written whilst in the School of Mathematics and Statistics at the University of St Andrews, was supervised by Prof Nikola Ruškuc, and was funded by EPSRC.

Research students

  • James Fraser (2016-) [second supervisor]
  • David Marchant (2016-) [part time]
  • Jakub Sliacan (2015-)
  • James Tuite (2015-) [second supervisor, part time]
  • David Bevan (2012-2015). [Thesis]


In Opinion 117, Doron Zeilberger (my academic grandfather!) suggested more people should upload grant bids, so here are mine.
  • [FUNDED] EPSRC First Grant, EP/J006130/1, £91,797, 1st October 2012 -- 30th September 2013. Infinite Antichains of Combinatorial Structures.
    [Proposal] [GoW]
  • [FUNDED] LMS Conference Grants (Scheme 1), £560, 30th January 2013. Winter Combinatorics Meeting, held at The Open University.
  • [NOT FUNDED] ERC Starting Grant, 2014. [Part B2]
  • [FUNDED] LMS Conference Grants (Scheme 1), £2,000, 11th--12th September 2014. Valediction to Jeremy Gray, held at Mercure Parkside Hotel, Milton Keynes.
  • [FUNDED] LMS Conference Grants (Scheme 1), £1,908, 15th--19th June 2015. Permutation Patterns 2015, held at De Morgan House, London.
  • [NOT FUNDED] EPSRC Standard Grant, 2015. [Proposal]

Conference Organisation

  • Local organising committee, Winter Combinatorics Meeting, Open University, annually 2011-2016.
  • Chair of Local Organising Committee, Permutation Patterns 2015, De Morgan House, London, 15-19 June 2015.
  • Organising Committee, Permutation Patterns 2011, California Polytechnic University, San Luis Obispo, USA, 20-24 June 2011.
  • Organising Committee, Permutation Patterns 2010, Dartmouth College, USA, 9-13 August 2010.
  • Local organising committee, Techniques and Problems in Graph Theory, University of Bristol, 1-3 July 2009.


Other Activities

  • Review for the LMS Newsletter of Alex's Adventures in Numberland by Alex Bellos. [PDF] [Newsletter]
    A version also appeared in Plus Magazine. [Plus]
  • Read my MathSciNet Reviews.
  • Referee for: Annals of Combinatorics, Discrete Applied Mathematics, Discrete Mathematics, Electronic Journal of Combinatorics, Graphs and Combinatorics, Journal of Combinatorial Theory Series A, Journal of Combinatorics, LMS Lecture Note Series, Mathematika, Pure Mathematics and Applications.

List of Coauthors


Submitted Papers

  1. Brignall, R., and Sliačan, J., Juxtaposing Catalan permutation classes with monotone ones.
    [PDF] [BibTeX] [Abstract]
  2. Brignall, R., Choi, H., Jeong, J., and Oum, S.-i., Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs.
    [PDF] [BibTeX] [Abstract]
  3. Albert, M.H., Brignall, R., Ruškuc, N., and Vatter, V., Rationality for subclasses of 321-avoiding permutations.
    [PDF] [BibTeX] [Abstract]
  4. Atminas, A., and Brignall, R., Well-quasi-ordering and finite distinguishing number.
    [PDF] [BibTeX] [Abstract]
  5. Atminas, A., Brignall, R., Lozin, V., and Stacho, J., Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs.
    [PDF] [BibTeX] [Abstract]

Refereed Publications

  1. Brignall, R., Korpelainen, N., and Vatter, V., Linear clique-width for hereditary classes of cographs. Journal of Graph Theory, accepted.
    [PDF] [Journal] [BibTeX] [Abstract]
  2. Albert, M.H., and Brignall, R., \(2\times 2\) monotone grid classes are finitely based. Discrete Mathematics and Theoretical Computer Science, 18(2), 2016, #1 (Permutation Patterns 2015)
    [PDF] [Journal] [BibTeX] [Abstract]
  3. Brignall, R., Lozin, V., and Stacho, J., Bichain graphs: geometric model and universal graphs. Discrete Applied Mathematics, 199 (2016), 16-29
    [PDF] [Journal] [BibTeX] [Abstract]
  4. Atminas, A., Brignall, R., Korpelainen, N., Lozin, V., and Vatter, V., Well-quasi-order for permutation graphs omitting a path and a clique. Electronic Journal of Combinatorics 22(2) (2015), #P2.20 (21pp)
    [PDF] [Journal] [BibTeX] [Abstract]
  5. Brignall, R., and Vatter, V., A simple proof of a theorem of Schmerl and Trotter for permutations. Journal of Combinatorics, 6 (2015) No. 1-2, 47-54
    [PDF] [Journal] [BibTeX] [Abstract]
  6. Albert, M.H., and Brignall, R., Enumerating indices of Schubert varieties defined by inclusions. Journal of Combinatorial Theory, Series A, 123 (2014), 154-168
    [PDF] [Journal] [BibTeX] [Abstract]
  7. Albert, M.H., Brignall, R., and Vatter, V., Large infinite antichains of permutations. Pure Mathematics and Applications, 24 (2013) No. 2, 47-57
    [PDF] [Journal] [BibTeX] [Abstract]
  8. Brignall, R., Georgiou, N., and Waters, R., Modular decomposition and the reconstruction conjecture. Journal of Combinatorics, 3 (2012), 123-134
    [PDF] [Journal] [BibTeX] [Abstract]
  9. Albert, M.H., Atkinson, M.D., and Brignall, R., The enumeration of three pattern classes using monotone grid classes. Electronic Journal of Combinatorics 19(3) (2012), #P20 (34pp)
    [PDF] [Journal] [BibTeX] [Abstract]
  10. Brignall, R., Grid classes and partial well order. Journal of Combinatorial Theory Series A, 119 (2012), 99-116
    [PDF] [Journal] [BibTeX] [Abstract]
  11. Albert, M.H., Atkinson, M.D., and Brignall, R., The enumeration of permutations avoiding 2143 and 4231. Pure Mathematics and Applications, 22 (2011) No. 2, 87-98
    [PDF] [Journal] [BibTeX] [Abstract]
  12. Brignall, R., Ruškuc, N., and Vatter, V., Simple extensions of combinatorial structures. Mathematika, 57 (2011), 193-214
    [PDF] [Journal] [BibTeX] [Abstract]
  13. Albert, M.H., Atkinson, M.D., Brignall, R., Ruškuc, N., Smith, R., and West, J., Growth rates for subclasses of Av(321). Electronic Journal of Combinatorics 17(1) (2010), #R141 (16pp)
    [PDF] [Journal] [BibTeX] [Abstract]
  14. Brignall, R., A survey of simple permutations. S. Linton, N. Ruškuc and V. Vatter, Eds., vol. 376 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 41-65
    [PDF] [Book] [BibTeX] [Abstract]
  15. Brignall, R., Ekhad, S.B., Smith, R., and Vatter, V., Almost avoiding permutations. Discrete Mathematics 309 (2009), 6626-6631
    [PDF] [Journal] [BibTeX] [Abstract]
  16. Brignall, R., Huczynska, S., and Vatter, V., Decomposing simple permutations, with enumerative consequences. Combinatorica, 28 (4) (2008) 385-400
    [PDF] [Journal] [BibTeX] [Abstract]
  17. Brignall, R., Huczynska, S., and Vatter, V., Simple permutations and algebraic generating functions. Journal of Combinatorial Theory Series A 115 (2008), 423-441
    [PDF] [Journal] [BibTeX] [Abstract]
  18. Brignall, R., Ruškuc, N., and Vatter, V., Simple permutations: decidability and unavoidable substructures. Theoretical Computer Science 391 (2008), 150-163
    [PDF] [Journal] [BibTeX] [Abstract]
  19. Albert, M.H., Atkinson, M.D., and Brignall, R., Permutation classes of polynomial growth. Annals of Combinatorics 11 (2007), no. 3-4, 249--264
    [PDF] [Journal] [BibTeX] [Abstract]
  20. Brignall, R., Wreath products of permutation classes. Electronic Journal of Combinatorics 14 (2007), #R46 (15pp)
    [PDF] [Journal] [BibTeX] [Abstract]


  • Brignall, R., Pattern classes of permutations: constructions, atomicity and the finite basis property, M.Sc. Thesis, September 2004. [PDF]
  • Brignall, R., Simplicity in Relational Structures and its Application to Permutation Classes, Ph.D. Thesis, October 2007. [PDF]


Invited Talks

Contributed Talks



School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA


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This page last modified on 07 November 2016 at 14:23. [Disclaimer]