Enumeration of totally positive Grassmann cells

TL;DR

This paper provides an explicit enumeration of cells in the totally positive Grassmannian, deriving a generating function, a new proof of its Euler characteristic, and a novel q-analog of Eulerian numbers.

Contribution

It introduces a generating function for counting Grassmannian cells by dimension and connects this to new q-analogs of classical combinatorial numbers.

Findings

Derived a generating function for cell enumeration in Gr_{kn}+

Proved the Euler characteristic of Gr_{kn}+ is 1

Developed a new q-analog of Eulerian numbers

Abstract

Alex Postnikov has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr_{kn}+, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of this paper is an explicit generating function which enumerates the cells in Gr_{kn}+ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Gr_{kn}+ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.