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Workshop

Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned inside-out polytopes

  • Sophie Rehberg (Freie Universität Berlin, Berlin, Germany)
Live Stream MPI für Mathematik in den Naturwissenschaften Leipzig (Live Stream)

Abstract

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley.

We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart--Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar--Ardila (2017) and Billera--Jia--Reiner (2009).

Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval--Karaboghossian--Tanasa (2020).

Links

conference
4/6/21 4/9/21

(Polytop)ics: Recent advances on polytopes

MPI für Mathematik in den Naturwissenschaften Leipzig Live Stream

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Federico Castillo

Max Planck Institute for Mathematics in the Sciences

Giulia Codenotti

Goethe University Frankfurt

Benjamin Schröter

Royal Institute of Technology (KTH)