双置换面体
The bipermutahedron
摘要：
调和多面体和双置换面体是两个相关的多面体出现在阿迪拉，德纳姆，和胡的工作拉格朗日几何拟阵。我们研究了双置换面体。我们证明了它的面与没有孤立顶点的顶点标记和边标记的多重图是双射的;它的f-向量的生成函数是三变量Rogers-Ramanujan函数的一个简单的估计。我们证明了双置换体扇的h-多项式是欧拉多项式，它根据双置换的下降数来计数双置换。我们构造了一个幺模三角剖分的产品n个三角形，这是组合等价于（三重锥）的第n个bipermutahedral风扇。Ehrhart理论然后给了我们一个公式的biEulerian多项式，我们用它来表明，这个多项式是实根和h-向量的双permutahedral风扇是对数凹和单峰。我们描述了所有的变形的双permutahedron，也就是说，充足的锥的双permutahedron复曲面品种。我们证明了在这个族中的所有多面体中，双置换面体具有最大可能的对称群。最后，我们证明了双置换面体与调和多面体的Minkowski商等于2。
英文摘要：
The harmonic polytope and the bipermutahedron are two related polytopes which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We study the bipermutahedron. We show that its faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its f-vector is a simple evaluation of the three variable Rogers--Ramanujan function. We show that the h-polynomial of the bipermutahedral fan is the biEulerian polynomial, which counts bipermutations according to their number of descents. We construct a unimodular triangulation of the product of n triangles that is combinatorially equivalent to (the triple cone over) the nth bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the h-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.
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