Non-hyperoctahedral categories of two-colored partitions part I: new categories

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