Brylawski proved the coefficients of the Tutte polynomial of a matroid
satisfy a set of linear relations. We extend these relations to a
generalization of the Tutte polynomial that includes greedoids and
antimatroids. This leads to families of new identities for antimatroids,
including trees, posets, chordal graphs and finite point sets in
$\mathbb{R}^n$. It also gives a "new" linear relation for matroids that is
implied by Brylawski's identities