Résumé
I will present six interrelated general expressions of the Tutte polynomial of a graph, that are available as soon as the set of edges is linearly ordered, and that witness combinatorial properties of such a graph:- the classical enumeration of spanning tree activities;- its refinement into a four variable expression in terms of subset activities (that corresponds to the classical partition of the set of edge subsets into boolean intervals);- the enumeration of orientation-activities for directed graphs;- its refinement into a four variable expression in terms of subset orientation-activities (that corresponds to the partition of the set of orientations into active partition reversal classes);- the convolution formula for the Tutte polynomial (that does not need the graph to be ordered);- and an expression of the Tutte polynomial using only beta invariants of minors (that refines the above expressions).I will mention that these expressions are all interrelated by the canonical active bijection between spanning trees and orientations, subject of a long-term joint work with Michel Las Vergnas.