Peter Cameron (Queen Mary, Univ London).

Orbit-counting for colourings, flows, and weight enumerators.

The Tutte polynomial of a matroid has many specialisations which count proper colourings or nowhere-zero flows of a graph, the weight enumerator of a linear code, etc. Given a group of automorphisms, we aim to produce a polynomial which has similar specialisations to count orbits of the group on colourings, flows, etc. Such a polynomial can be defined for certain matroids (those representable over principal ideal domains). Many questions about its properties are still open.