二分划的偏序集
The poset of bipartitions
摘要：
Foata和Zeilberger在描述导致相关反演统计量和主指数等分布的关系时引入了二分关系。我们考虑由包含给出的二分关系上的自然偏序。我们表明，相对于这个偏序，一组大小$n$的二分关系形成一个等级格的秩$3n-2$。证明了该格的序复形同伦等价于一个$n-2$维的球面.该格中的每一真区间要么有可缩序复形，要么同构于布尔格与二分关系的小格的直积，从而得到每一真区间的默比乌斯函数为0，1或-1.在证明的主要工具是离散莫尔斯理论发展的福尔曼，并应用这一理论，以配合物的分次偏序集，设计的巴布森和赫什，在扩展形式的赫什和韦尔克。
英文摘要：
Bipartitional relations were introduced by Foata and Zeilberger in their characterization of relations which give rise to equidistribution of the associated inversion statistic and major index. We consider the natural partial order on bipartitional relations given by inclusion. We show that, with respect to this partial order, the bipartitional relations on a set of size $n$ form a graded lattice of rank $3n-2$. Moreover, we prove that the order complex of this lattice is homotopy equivalent to a sphere of dimension $n-2$. Each proper interval in this lattice has either a contractible order complex, or it is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations.As a consequence, we obtain that the Möbius function of every interval is 0, 1, or -1. The main tool in the proofs is discrete Morse theory as developed by Forman, and an application of this theory to order complexes of graded posets, designed by Babson and Hersh, in the extended form of Hersh and Welker.
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