Séminaire Lotharingien de Combinatoire, 89B.43 (2023), 11 pp.

Matthias Beck, Danai Deligeorgaki, Max Hlavacek and Jerónimo Valencia-Porras

Inequalities for f*-vectors of Lattice Polytopes

Abstract. The Ehrhart polynomial ehrP(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f*-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr<P(n) with respect to the binomial coefficient basis {(n-1)(n-1) (n-1)} 0 , 1 ,..., d, where d = dim P. Similarly to h/h*-vectors, the f*-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f*-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal.

Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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