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                <term xml:lang="en">permutahedra</term>
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                <term xml:lang="en">associahedra</term>
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                <term xml:lang="fr">Permutoèdre</term>
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              <p>We introduce the shuffle of deformed permutahedra (a.k.a. generalized permutahedra), a simple associative operation obtained as the Cartesian product followed by the Minkowski sum with the graphical zonotope of a complete bipartite graph. Besides preserving the class of graphical zonotopes (the shuffle of two graphical zonotopes is the graphical zonotope of the join of the graphs), this operation is particularly relevant when applied to the classical permutahedra and associahedra. On the one hand, the shuffle of an $m$-permutahedron with an $n$-associahedron gives the $(m,n)$-multiplihedron, whose face structure is encoded by $m$-painted $n$-trees, generalizing the classical multiplihedron. We show in particular that the graph of the $(m,n)$-multiplihedron is the Hasse diagram of a lattice generalizing the weak order on permutations and the Tamari lattice on binary trees. On the other hand, the shuffle of an $m$-anti-associahedron with an $n$-associahedron gives the $(m,n)$-biassociahedron, whose face structure is encoded by $(m,n)$-bitrees, with relevant connections to bialgebras up to homotopy. We provide explicit vertex, facet, and Minkowski sum descriptions of these polytopes, as well as summation formulas for their $f$-polynomials based on generating functionology of decorated trees.</p>
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              <p>On introduit une opération associative de mélange des permutoèdres déformés (aussi connus sous le nom de permutoèdres généralisés), obtenue comme le produit cartésien suivi de la somme de Minkowski avec le zonotope graphique d'un graphe bipartite complet. Outre le fait qu'elle préserve la classe des zonotopes graphiques, cette opération est spécialement intéressante lorsqu'elle est appliquée aux permutoèdres et aux associaèdres. Le mélange d'un m-permutoèdre et d'un n-associaèdre donne le (m,n)-multiplièdre, dont les faces sont décrites par des arbres peints. Le mélange d'un associaèdre et d'un anti-associaèdre donne un bi-associaèdre, famille de polytopes liée aux bigèbres à homotopie près. On donne des descriptions explicites des sommets et facettes de ces polytopes, ainsi que des formules pour leurs f-polynômes.</p>
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