**Faculty**-
Dr David Bevan

Dr Sergey Kitaev

Prof Einar Steingrímsson

**Honorary Senior Lecturer**-
Dr Mark Dukes

**Research Associates**-
Dr Thomas Selig

Dr Jason Smith **PhD Students**- Marc Glen

The group's research interests are in enumerative, bijective, algebraic and topological combinatorics, with connections to theoretical computer science, physics and graph theory. Much of our work has been connected to permutation patterns, a relatively young but very active research area, with several hundred papers published in the last ten years and several distinct lines of research emerging lately.

A recurring theme in much of our work is to find connections between families of different combinatorial structures, in the form of bijections that send a set of statistics on one side to a set of statistics on the other. Such statistics-preserving bijections often reveal previously unknown properties of the structures being studied.

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Here are some examples of our recent work, with links to papers:

Families of combinatorial objects often have a natural partial order on them, and such posets always have a corresponding simplicial complex, the order complex, associated to them. Classical questions for these complexes are about their topological properties, in particular the Möbius function, which is a topological invariant. We have lately been studying the topology and Möbius function of intervals in the poset of permutations, ordered by pattern containment.

Since their introduction by Fishburn in 1970, interval orders, or (2+2)-free posets, have received a lot of attention. Their enumeration resisted all attempts until 2010 when we found an encoding of them as ascent sequences. These sequences also encode permutations avoiding a certain bivincular pattern. We have more recently discovered an intriguing representation of labeled interval orders as pairs of permutations.

Many models in theoretical physics have a combinatorial flavour. Our recent work in this area includes new results on the abelian sandpile model and the introduction and analysis of new combinatorial structures called web worlds and web diagrams that are at the crossroads of graph theory and order theory. The motivation for these new structures comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge theories.

The field of word-representable graphs, having its roots in algebra, is a new area of discrete mathematics that lies on the boundary of graph theory and combinatorics on words. These graphs are a generalization of several classical classes of graphs and they are related to theoretical computer science and scheduling problems. The study of word-representable graphs is initiated in our paper, while a fundamental result in the area, a characterization theorem for such graphs in terms of certain graph orientations, is obtained in this paper.

- EPSRC grant: New Combinatorial Perspectives on the Abelian Sandpile model

Principal Investigator: Mark Dukes

Amount: £ 354,000

Funding period: 2015 to 2018

- EPSRC grant: The Möbius function of the poset of permutations

Principal Investigator: Einar Steingrímsson

Amount: £ 356,000

Funding period: 2015 to 2018

For papers and preprints please visit the individual members' websites.