An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

A bit long for a comment.

I like this question because as far as I know there are a lot of open problems concerned with expressing the links you mentioned via combinatorics. Perhaps you'd be interested in the following: "On some combinatorial and algebraic properties of Dunkl elements" - Anatol Kirillov.

The basic idea in that paper is that certain specializations and multiparameter deformations of Schubert and Grothendieck polynomials can be expressed through Narayana numbers, Catalan-Hankel determinants, and Schroder numbers, including some generalizations as well. In addition, these specializations come up in other areas such as $k$-triangulations of convex polygons and Carlitz-Riordan numbers. Through the wonderous abstract nonsense of Flags and degenerate loci, you can perhaps glean some knowledge.

In this recent preprint of Postnikov https://arxiv.org/abs/1806.05307 (which is for an upcoming ICM talk in Rio) he explains a beautiful direct connection between the geometry of the positive Grassmannian and the combinatorics of hypersimplices (polytopes whose normalized volume are the Eulerian numbers).