Enumeration of totally positive Grassmann cells
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.
Keywords
Cite
@article{arxiv.math/0307271,
title = {Enumeration of totally positive Grassmann cells},
author = {Lauren K. Williams},
journal= {arXiv preprint arXiv:math/0307271},
year = {2007}
}
Comments
21 pages, 10 figures. Final version, with references added and minor corrections made, to appear in Advances in Mathematics