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

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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