In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra. The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the h*-polynomial, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the h*-polynomial are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegativ

Vortrag im Stochastic Afternoon: https://www.sfb1283.uni-bielefeld.de/talks/view/17586

Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Marina Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory. We will then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these in the appropriate association scheme using Delsarte Theory. No particular knowledge of association schemes will be required to appreciate this talk

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