Permutahedra and associahedra are classical structures at the interface between combinatorics, discrete geometry, algebra, and algorithms. I will first survey some of their properties and then present the generalization of quotientopes, a family of polytopes realizing any lattice quotient of the weak order on permutations. The talk is based on works of N. Reading and on joint collaborations with F. Santos, A. Padrol, and J. Ritter.
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