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Classification of combinatorial polynomials (in particular, Ehrhart polynomials of zonotopes)

Positivity Questions in Geometric Combinatorics July 10, 2017 - July 21, 2017

July 19, 2017 (10:30 AM PDT - 11:30 AM PDT)
Speaker(s): Matthias Beck (San Francisco State University)
Location: MSRI: Simons Auditorium
Video

Abstract

The Ehrhart polynomial of a lattice polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic polynomials?, we give a brief survey of attempts during the last half century to classify Ehrhart polynomials. It turns out that this classification problem is related to that of a whole family of polynomials in combinatorics.

 

We will present some new results for Ehrhart polynomials of zonotopes, i.e., projections of (higher dimensional) cubes. This includes a combinatorial description in terms of refined descent statistics of permutations and a formula in matroidal terms which complements a well-known zonotopal identity of Stanley (1991). Finally, we give a complete description of the convex hull of the Ehrhart coefficients of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

 

New results in this talk comes from joint work with Katharina Jochemko (KTH) and Emily McCullough (University of San Francisco).

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22-Beck

H.264 Video 22-Beck.mp4 153 MB video/mp4 rtsp://videos.msri.org/data/000/029/010/original/22-Beck.mp4 Download
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