Counting weighted maximal chains in the circular Bruhat order

TL;DR

This paper derives a closed-form formula for summing weighted chains in the circular Bruhat order, which relates to the face poset of the totally nonnegative Grassmannian, advancing combinatorial understanding of this geometric structure.

Contribution

It introduces a novel closed formula for weighted chain sums in the circular Bruhat order, connecting combinatorics with geometric cell decompositions.

Findings

Provides a closed formula for weighted chain sums

Links combinatorial chains to geometric cell structures

Enhances understanding of the positroid cell decomposition

Abstract

The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Pl\"ucker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).