Williams, Lauren2009-02-202005Williams, Lauren K. 2005. Enumeration of totally positive Grassmann cells. Advances in Mathematics 190(2): 319-342.0001-8708http://nrs.harvard.edu/urn-3:HUL.InstRepos:2624454Postnikov (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.en-USGrassmannianQ-analogsEulerian numberstotal positivity10.1016/j.aim.2004.01.003