Research
My research interests broadly lie in the study of combinatorial and relational structures. I have paid particular attention to the study of permutation containment and avoidance, but I am also interested in the connections between this area and other classes of relational structures -- for example hereditary properties of graphs, tournaments and posets. In particular I have been interested in the so-called simple (or prime, or indecomposable) structures which form the elemental "building blocks" of the substitution decomposition (or modular decomposition). For permutations these simple structures are particularly important, and their presence (or otherwise) in permutation classes have multiple far-reaching consequences.
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 (Reykjavík 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.
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 (Reykjavík 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.
Submitted Papers
Refereed Publications
- 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, accepted. [PDF] [BibTeX] - Brignall, R., Ruškuc, N.,
and Vatter,
V., Simple extensions of combinatorial structures.
Mathematika, accepted. [PDF] [BibTeX] - Brignall, R., A survey of simple permutations.
In Permutation Pattern (2010), 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] - Brignall, R., Ekhad, S., Smith, R.,
and Vatter,
V., Almost avoiding permutations.
Discrete Mathematics 309 (2009), 6626-6631. [PDF] [Journal] [BibTeX] - Brignall, R., Huczynska, S.,
and Vatter,
V., Decomposing simple permutations, with enumerative consequences.
Combinatorica, 28 (4) (2008) 385-400. [PDF] [Journal] [BibTeX] - 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] - Brignall, R., Ruškuc, N.,
and Vatter,
V., Simple permutations: decidability and unavoidable substructures.
Theoretical Computer Science 391 (2008), 150-163. [PDF] [Journal] [BibTeX] - 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] - Brignall, R., Wreath products of permutation classes.
Electronic Journal of Combinatorics 14 (2007), #R46 (15pp). [PDF] [Journal] [BibTeX]
Theses
- 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]
Talks
- 8th International Conference on Permutation Patterns, Dartmouth College, NH, USA, 10th August 2010. [Notes - 1.2MB]
- From A=B to Z=60, Conference in Honor of Doron Zeilberger's 60th Birthday, Rutgers University, NJ, USA, 28th May 2010. [Abstract] [Video]
- Pure Mathematics Colloquium, St Andrews, Scotland, 13th May 2010. [PDF]
- Combinatorics Seminar, Dartmouth College, NH, USA, 8th February 2010. [PDF]
- DIMAP Seminar, Warwick, UK, 3rd November 2009. [PDF]
- 7th International Conference on Permutation Patterns, Florence, Italy, 15th July 2009. [PDF]
- 22nd British Combinatorial Conference, St Andrews, Scotland, 7th July 2009. [PDF]
- 6th International Conference on Permutation Patterns, Dunedin, New Zealand, 17th June 2008. [PDF]
- Algebra and Geometry Seminar, Bristol, UK, 13th February 2008 [PDF]
- 32nd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, Dunedin, New Zealand, 7th December 2007 [PDF]
- 5th International Conference on Permutation Patterns, St Andrews, Scotland, 13th June 2007. [PDF]
- 18th Postgraduate Combinatorial Conference, St Andrews, Scotland, 6th June 2007. [PDF]
- 4th International Conference on Permutation Patterns, Reykjavik, Iceland, 15th June 2006. [PDF]
- EMS Postgraduate Students' Meeting, The Burn, Edzell, 11th May 2005. A shorter and less technical version of the above talk. [PDF]
- 3rd International Conference on Permutation Patterns, Gainesville, Florida, USA, 10th March 2005. [PDF]
