The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations

We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientations in graphs. In corollary, we obtain expressions for the partial derivatives of the Tutte polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table

The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities

We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.Comment: 28 pages, 3 figures, 5 table

Matroid Chern-Schwartz-MacPherson cycles and Tutte activities

Lop\'ez de Medrano-Rin\'con-Shaw defined Chern-Schwartz-MacPherson cycles for an arbitrary matroid $M$ and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of $T(M;x,0)$, where $T(M;x,y)$ is the Tutte polynomial associated to $M$. Ardila-Denham-Huh recently utilized this interpretation of these coefficients in order to demonstrate their log-concavity. In this note we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appropriate codimension and showing that the weighted point count agrees with the Gioan-Las Vergnas refined activities expansion of the Tutte polynomial

Fourientation activities and the Tutte polynomial

International audienceA fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of G which allow for a new 12 variable expansion of the Tutte polynomial TG. Our formula specializes to both an orientation activities expansion of TG due to Las Vergnas (1984) and a generalized activities expansion of TG due to Gordon and Traldi (1990)