变形置换多面体、乘积多面体、约束多面体与双结合多面体的洗牌
Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra
AI总结：
引入变形置换多面体的洗牌运算，通过笛卡尔积与Minkowski和生成乘积多面体、约束多面体和双结合多面体，并给出其面结构编码、格论性质及显式几何描述。
摘要：
我们引入了变形置换多面体（又称广义置换多面体）的洗牌操作，这是一种简单的结合运算，通过先进行笛卡尔积，再与完全二分图的图形 zonotope 进行 Minkowski 和得到。除了能保持图形 zonotope 类（两个图形 zonotope 的洗牌是其图连接的图形 zonotope），该运算应用于经典置换多面体和结合多面体时尤为相关。首先，$m$-置换多面体与 $n$-结合多面体的洗牌给出 $(m,n)$-乘积多面体，其面结构由 $m$-涂色 $n$-树编码，推广了经典乘积多面体。我们特别证明，$(m,n)$-乘积多面体的图是推广了排列弱序和二叉树 Tamari 格的格的 Hasse 图。其次，$m$-结合多面体与 $n$-结合多面体的洗牌给出 $(m,n)$-约束多面体，其面结构由 $(m,n)$-余树编码，反映了受网格约束的粒子碰撞。第三，$m$-反结合多面体与 $n$-结合多面体的洗牌给出 $(m,n)$-双结合多面体，其面结构由 $(m,n)$-双树编码，与同伦双代数有重要联系。我们提供了这些多面体的显式顶点、面和 Minkowski 和描述，以及基于装饰树生成函数论的 $f$-多项式求和公式。
英文摘要：
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.
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