Schubert matroids, Delannoy paths, and Speyer's invariant
arXiv:2311.01397 · doi:10.5070/C63362796
Abstract
We provide a combinatorial way of computing Speyer's -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 -polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the -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 -polynomial of an arbitrary matroid.
20 pages. 1 ancillary file. To appear in Combinatorial Theory