Counting weighted maximal chains in the circular Bruhat order

Abstract

The totally nonnegative Grassmannian $\mathrm{Gr}(k,n{)}_{\ge 0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n{)}_{\ge 0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).

8 pages