Classification of combinatorial polynomials (in particular, Ehrhart polynomials of zonotopes)
Positivity Questions in Geometric Combinatorics July 10, 2017  July 21, 2017
Location: SLMath: Eisenbud Auditorium
22Beck
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 muchcherished and stillwideopen 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 wellknown 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).
22Beck
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