Mathematics – Combinatorics
Scientific paper
2003-07-20
Mathematics
Combinatorics
21 pages, 10 figures. Final version, with references added and minor corrections made, to appear in Advances in Mathematics
Scientific paper
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.
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