A Greedoid Polynomial Which Distinguishes Rooted Arborescences

We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of fG(t, z). We prove (Theorem 2.8) that when G is a rooted directed arborescence, fo(t, z) completely determines the arborescence. We also show the polynomial is irreducible over Z[t, z] for arborescences with only one edge directed out of the distinguished vertex. When G is a matroid, fc(t, z) coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows fa(t, z) does not distinguish rooted arborescences among the class of all greedoids.