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 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.

Foata和Zeilberger在描述关系时引入了二元关系，这引起了相关的反演统计量和主要指数的平均分配。我们考虑了包含关系给定的二元关系的自然偏序。我们表明，关于这个偏序，在一组大小上的两分关系$ñ$ 形成等级的等级格 $3ñ-\mathrm{2个}$。此外，我们证明了该晶格的阶复杂是同构的，等价于一个维数球$ñ-\mathrm{2个}$。该晶格中的每个适当间隔要么具有可收缩阶数，要么与布尔晶格和较小二元关系晶格的直接乘积同构。结果，我们得出每个间隔的Möbius函数为0、1或-1。证明中的主要工具是福尔曼（Forman）开发的离散摩尔斯理论，并将该理论应用到由巴布森（Babson）和赫什（Hersh）设计的以扩展的赫什（Hersh）和韦尔克（Welker）的形式定序的梯度波状体的复合物中。