We introduce a "lifting'' construction for generalized permutohedra, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra,'' answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial'' arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal.

Source : oai:HAL:hal-01283117v1

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)

Section: Proceedings

Published on: January 1, 2012

Submitted on: January 31, 2017

Keywords: permutohedron, associahedron, multiplihedron, nestohedron, subdivision, composition polynomial, polynomial interpolation,Polytope,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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