Two-colored partitions are the objects at the heart of the combinatorial description of the co-representation categories of a particular class of compact matrix quantum groups, the so-called unitary easy quantum groups, via categories of two-colored partitions.
We say that is a two-colored partition if there exist two totally ordered finite (not necessarily non-empty) sets , the lower row, and , the upper row of , a set-theoretical partition of a disjoint union of and and a mapping , the coloring of , such that .
The set of all two-colored partitions is denoted by .
The elements of are called lower points and those of upper points. And for all we write for the set of all two-colored partitions with upper and lower points.