We define a two-variable polynomial ${f_G}(t,z)$ for a greedoid $G$ which generalizes the standard one-variable greedoid polynomial ${\lambda _G}(t)$. Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of ${f_G}(t,z)$. We prove (Theorem 2.8) that when $G$ is a rooted directed arborescence, ${f_G}(t,z)$ completely determines the arborescence. We also show the polynomial is irreducible over ${\mathbf {Z}}[t,z]$ for arborescences with only one edge directed out of the distinguished vertex. When $G$ is a matroid, ${f_G}(t,z)$ coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows ${f_G}(t,z)$ does not distinguish rooted arborescences among the class of all greedoids. References
T. Brylawski and D. Kelly, Matroids and combinatorial geometries, Carolina Lecture Series, University of North Carolina, Department of Mathematics, Chapel Hill, N.C., 1980. MR 573268