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Compact quantum groups can be studied by investigating theirco-representation categories in analogy to the Schur-Weyl/Tannaka-Kreinapproach. For the special class of (unitary) "easy" quantum groups thesecategories arise from a combinatorial structure: Rows of two-colored pointsform the objects, partitions of two such rows the morphisms;vertical/horizontal concatenation and reflection give composition, monoidalproduct and involution. Of the four possible classes $\mathcal{O}$,$\mathcal{B}$, $\mathcal{S}$ and $\mathcal{H}$ of such categories (inspiredrespectively by the classical orthogonal, bistochastic, symmetric andhyperoctahedral groups) we treat the first three -- the non-hyperoctahedralones. We introduce many new examples of such categories. They are defined interms of subtle combinations of block size, coloring and non-crossingconditions. This article is part of an effort to classify allnon-hyperoctahedral categories of two-colored partitions. The article is purelycombinatorial in nature; The quantum group aspects are left out.

Non-Hyperoctahedral Categories of Two-Colored Partitions, Part I: New Categories