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# f^*-vectors of lattice polytopes (Max Hlavacek, Pomona College)

## November 14, 2023 @ 12:15 pm - 1:10 pm

The Ehrhart polynomial of a lattice polytope P counts the number of integer points in the nth integral dilate of P. The f^* -vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of the Ehrhart polynomial with respect to the binomial coefficient basis . Similarly to h and h^* -vectors, the f^* -vector of P coincides with the f-vector (counting faces of every dimension) of its unimodular triangulations (if they exist). We give 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. Even though f^* -vectors of polytopes are not always unimodal, we describe several families of polytopes that carry the unimodality property.