Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats

Abstract

We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize similiar statements for perfect matroid designs and near designs due to Brylawski (1980) and help to understand families of matroids with identical Tutte polynomials as constructed by Giménez and later improved by Shoda (2012).