Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra

TL;DR

This paper introduces a new shuffle operation on deformed permutahedra that generates complex polytopes like multiplihedra, constrainahedra, and biassociahedra, revealing their combinatorial structures and algebraic connections.

Contribution

It defines the shuffle operation on generalized permutahedra and characterizes the resulting polytopes' face structures and combinatorial properties.

Findings

The shuffle of permutahedra and associahedra yields multiplihedra, constrainahedra, and biassociahedra.

The face structures are encoded by painted trees, cotrees, and bitrees.

Explicit descriptions and formulas for vertices, facets, and f-polynomials are provided.

Abstract

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. First, 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. Second, the shuffle of an $m$-associahedron with an $n$-associahedron gives the $(m,n)$-constrainahedron, whose face structure is encoded by $(m,n)$-cotrees, and reflects collisions of particles constrained on a grid. Third, 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.