  We classify certain categories of partitions of finite sets subject to
specific rules on the colorization of points and the sizes of blocks. More
precisely, we consider pair partitions such that each block contains exactly
one white and one black point when rotated to one line; however crossings are
allowed. There are two families of such categories, the first of which is
indexed by cyclic groups and is covered in the present article; the second
family will be the content of a follow-up article. Via a Tannaka-Krein result,
the categories in the two families correspond to easy quantum groups
interpolating the classical unitary group and Wang's free unitary quantum
group. In fact, they are all half-liberated in some sense and our results imply
that there are many more half-liberation procedures than previously expected.
However, we focus on a purely combinatorial approach leaving quantum group
aspects aside.
