International audience We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra. On exprime le polytope matroïde $P_M$ d'un matroïde $M$ comme somme signée de Minkowski de simplices, et on obtient une formule pour le volume de $P_M$. Ceci donne une expression combinatoire pour le degré d'une clôture d'orbite de tore dans la Grassmannienne $Gr_{k,n}$. Ensuite, on déduit des résultats analogues pour le polytope ensemble indépendant et pour le polytope matroïde drapeau associé à $M$. Nos preuves sont fondées sur une extension naturelle de la théorie de Postnikov de permutoèdres généralisés.

We express the matroid polytope P M of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P M . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr k,n . We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.

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. We introduce a lifting construction for these polytopes, 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 present evidence suggesting that they may also be unimodal.

International audience 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.

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