A Characterization of Tutte Invariants of 2-Polymatroids

This paper develops a theory of Tutte invariants for 2-polymatroids that parallels the corresponding theory for matroids. It is shown that such 2-polymatroid Invariants arise in the enumeration of a wide variety of combinatorial structures including matchings and perfect matchings in graphs, weak colourings in hypergraphs, and common bases in pairs of matroids. The main result characterizes all such invariants proving that, with some trivial exceptions, every 2-polymatroid Tutte invariant can be easily expressed in terms of a certain two-variable polynomial that is closely related to the Tutte polynomial of a matroid.