A linking polynomial of two matroids

Abstract

We introduce a 4-variable polynomial Q as a natural invariant linking a pair of matroids on a common ground set. The polynomial Q has similarities with the classical Tutte polynomial of a single matroid, it contains as specialisations the generating function of common independent and spanning sets of a given size, it behaves naturally under a duality transform and there is a recipe theorem which shows that essentially it is the unique invariant satisfying simultaneous delete/contract recursions on a pair of matroids.