Transitive and Gallai colorings of the complete graph

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of incomparability graphs and anti-Ramsey theory. A directed analogue, called transitive coloring, was introduced by Berenstein, Greenstein and Li in a rather general setting. It is studied here for the acyclic tournament. The interplay of the two notions yields new enumerative results and algebraic perspectives. We first count Gallai and transitive colorings of the complete graph which use the maximal number of colors. The quasisymmetric generating functions of these colorings, equipped with a natural descent set, are shown to be Schur-positive for any number of colors. Explicit Schur expansions are described when the number of colors is maximal. It follows that descent sets of maximal Gallai and transitive colorings are equidistributed with descent sets of perfect matchings and pattern-avoiding indecomposable permutations, respectively. Corresponding commutative algebras are also studied. Their dimensions are shown to be equal to the number of Gallai colorings of the complete graph and the number of transitive colorings of the acyclic tournament, respectively. Relations to Orlik-Terao algebras are established.