Invariants of Tutte partitions and a q-analogue

We describe a construction of the Tutte polynomial for both matroids and q-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct q-analogue of such partitions. We propose axioms of a q-Tutte-Grothendieck invariant and show that this yields a q-analogue of a Tutte-Grothendieck invariant. We establish the connection between the rank generating polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.