The Active Bijection 2.a - Decomposition of activities for matroid bases, and Tutte polynomial of a matroid in terms of beta invariants of minors

Abstract

We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the active closure. It relies only on the fundamental bipartite graph of the basis and can be expressed also as a decomposition of general bipartite graphs on a linearly ordered set of vertices. From this, first, structurally, we obtain that the set of all bases can be canonically partitioned and decomposed in terms of such bases of minors induced by filtrations. Second, enumeratively, we derive an expression of the Tutte polynomial of a matroid in terms of beta invariants of minors. This expression refines at the same time the classical expressions in terms of basis activities and orientation activities (if the matroid is oriented), and the well-known convolution formula for the Tutte polynomial. Third, in a companion paper of the same series (No. 2.b), we use this decomposition of matroid bases, along with a similar decomposition of oriented matroids, and along with a bijection in the 1/0 activity case from a previous paper (No. 1), to define the canonical active bijection between orientations/signatures/reorientations and spanning trees/simplices/bases of a graph/real hyperplane arrangement/oriented matroid, as well as various related bijections.