Enumeration of totally positive Grassmann cells

Lauren K. Williams


February 13,
refreshments at 3:45pm


Alex Postnikov recently gave 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.

Speaker's Contact Info: lauren(at-sign)math.mit.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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