Polymake allowed me to create a "matroid" using {{0,1}, {1,3}, {2,3}} as my set of bases on [4], which is not surprising since I know that polymake does not automatically check if the sets given actually form the bases of a matroid.
However, it also tells me that this "matroid" has Tutte polynomial x^2+xy^2+xy. What exactly is it computing?
