Lifted generalized permutahedra and composition polynomials

Federico Ardila,Jeffrey Doker

doi:10.46298/dmtcs.3094

Abstract

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. Nous introduisons une construction de "lifting'' (redressement) pour permutaèdres généralisés, qui transforme un permutaèdre généralisé de dimension $n$ en un de dimension $n+1$. Nous démontrons que cette construction conduit au multiplièdre de Stasheff à partir de la théorie d'homotopie, et aux "nestomultiplièdres", ce qui répond à deux questions de Devadoss et Forcey. Nous construisons une subdivision de n'importe quel permutaèdre généralisé dont les pièces sont indexées par compositions. La volume de chaque pièce est donnée par un polynôme dont nous recherchons les propriétés combinatoires. Nous montrons comment ce "polynôme de composition'' surgit naturellement dans l'interpolation d'une fonction exponentielle. Nous démontrons que ses coefficients sont strictement positifs, et nous conjecturons qu'ils sont unimodaux.

Highlights

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way
These polytopes, closely related to polymatroids and recently re-introduced by Postnikov [15] have been the subject of great attention due their very rich combinatorial structure
Federico Ardila and Jeffrey Doker polytopes which naturally appear in homotopy theory, in geometric group theory, and in various moduli spaces: permutahedra, matroid polytopes [2], Pitman-Stanley polytopes [14], Stasheff’s associahedra [23], Carr and Devadoss’s graph associahedra [4], Stasheff’s multiplihedra [23], Devadoss and Forcey’s multiplihedra [6], and Feichtner and Sturmfels’s and Postnikov’s nestohedra [15, 8]

Summary

Introduction

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. Lifting P (q) permutahedron Pn+1 multiplihedron Jn graph multiplihedron J G nestomultiplihedron J B independent set polytope IM Definition 2.1 Given a generalized permutahedron P = Pn({zI }) in Rn and a number 0 ≤ q ≤ 1, define the q-lifting of P to be the polytope P (q) given by the inequalities n+1 ti = z[n], i=1 ti ≥ qzI for I ⊆ [n], i∈I

Results

Conclusion