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"][1] - Anatol Kirillov. 

The basic idea in that paper is that certain specializations and multiparamter 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 Flagss and degenerate loci, you can perhaps glean some knowledge.

  [1]: http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1746.pdf