Combinatorial Theory

We provide a combinatorial way of computing Speyer's \(g\)-polynomial on arbitrary Schubert matroids via the enumeration of certain Delannoy paths. We define a new statistic of a basis in a matroid, and express the \(g\)-polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the \(g\)-polynomial of Schubert matroids are deduced from our expression. Finally, we combine our formulas with a fundamental result of Derksen and Fink to provide an algorithm for computing the \(g\)-polynomial of an arbitrary matroid.

Mathematics Subject Classifications: 05B35, 52B40, 14T15

Keywords: Schubert matroids, \(g\)-polynomial, matroid polytopes, series-parallel matroids, lattice path enumeration