  Compact quantum groups can be studied by investigating their
co-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. The article is purely
combinatorial in nature; The quantum group aspects are left out.
