Strong Tutte functions of matroids and graphs

A strong Tutte function of matroids is a function of finite matroids which satisfies F(M 1 ○+M 2 ) = F(M 1 )F(M 2 ) and F(M)=a e F(M/e)+ b e F(M/e) for e not a loop or coloop of M, where a e , b e are scalar parameters depending only on e. We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial