Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the special class of (unitary) “easy” quantum groups, these categories arise from a combinatorial structure: rows of two-colored points form the objects, partitions of two such rows the morphisms. Vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes \({\mathcal {O}}\), \({\mathcal {B}}\), \({\mathcal {S}}\) and \({\mathcal {H}}\) of such categories (inspired, respectively, by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups), we treat the first three—the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. It is purely combinatorial in nature. The quantum group aspects are left out.

可以通过类似于Schur-Weyl / Tannaka-Krein方法研究其表示类别来研究紧密量子群。对于（单一）“易”量子组的特殊类别，这些类别源自组合结构：由两色点组成的行形成对象，由两行这样的行态构成。垂直/水平级联和反射产生成分，单项积和对合。在四个可能的类\（{\ mathcal {O}} \），\（{\ mathcal {B}} \），\（{\ mathcal {S}} \}和\（{\ mathcal {H}} \）在这些类别中（分别受经典的正交，双随机，对称和超八面体组的启发），我们对待前三个（非八面体组）进行处理。我们介绍了此类的许多新示例。它们是根据块大小，着色和非交叉条件的微妙组合定义的。本文是对所有两种颜色分区的非超八面体类别进行分类的工作的一部分。它本质上纯粹是组合的。省略了量子组方面。