Enumeration of $2$-Polymatroids on up to Seven Elements

Abstract

A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of $2$-polymatroids, up to isomorphism. When we implemented this algorithm on a computer, obtaining all $2$-polymatroids on at most seven elements, we discovered the surprising fact that the number of $2$-polymatroids on seven elements fails to be unimodal in rank.