The Arboricity Polynomial

Abstract

We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the counting function as a special case of a scheduling polynomial, i.e. both in terms of quasisymmetric functions and via Ehrhart theory of the normal fan of the matroid base polytope. We show basic properties of the polynomial and demonstrate that it is not a Tutte invariant. Namely, the arboricity polynomial does not satisfy a contraction / deletion recursion.