论文标题
变形的定居者，多皮希德拉，约束和偏见
Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra
论文作者
论文摘要
我们介绍了变形的Permutahedra（又称广义Permutahedra）的混乱，这是一个简单的关联操作，作为笛卡尔产物获得的，然后是Minkowski Sum，其图形界限具有完整的两极图的图形扎根。除了保留一类图形地位型（两个图形地图的洗牌是图形连接的图形地位）外，当应用于经典的Persutial Permutahedra and Associahedra时，此操作尤其重要。首先，带有$ n $ n $ associahedron的$ M $ -PERMUTAHEREDRON的$（m，n）$ - 多人二皮德隆，其面部结构由$ m $ painted $ n $ -trees编码，从而概括了经典的多人型。我们特别表明，$（m，n）$ - 多平面的图形是晶格的Hasse图，该晶格概括了排列上的弱顺序和二进制树上的tamari晶格。其次，带有$ n $ n $ - associahedron的$ M $ -MOSSOCIAHEDRON的shuffle给出了$（m，n）$ - 约束，其面部结构由$（m，n）$ - cotrees编码，反映了粒子上约束网格的碰撞。第三，带有$ n $ n $ -sassociahedron的$ M $ -MANTI-ASSOCIAHEDRON的混乱给出了$（m，n）$ - biassociahedron，其脸部结构由$（m，n）$ - bitrees编码，与Bialgebras相关的连接到同型。我们为这些多面体提供明确的顶点，刻面和Minkowski总和描述，以及基于装饰树的生成功能学的$ f $ polynomials的求和公式。
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.
