# Enumeration of Totally Positive Grassmann Cells

 Title: Enumeration of Totally Positive Grassmann Cells Author: Williams, Lauren Citation: Williams, Lauren K. 2005. Enumeration of totally positive Grassmann cells. Advances in Mathematics 190(2): 319-342. Full Text & Related Files: Williams_Enumeration.pdf (362.4Kb; PDF) Abstract: Postnikov (Webs in totally positive Grassmann cells, in preparation) 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 our work 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. Published Version: http://dx.doi.org/10.1016/j.aim.2004.01.003 Other Sources: http://arxiv.org/pdf/math/0307271v1 http://www.math.harvard.edu/~lauren/TNNsubmit.ps Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2624454 Downloads of this work: