##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## K. Williams

#### MIT Graduate Student

## Enumeration of totally positive Grassmann cells

##### Abstract:

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.

Host: Jeff Remmel

### April 27, 2004

### 4:00 PM

### AP&M 7321

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