Two-Colored Partition

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.

Definition

We say that $p$ is a two-colored partition if there exist two totally ordered finite (not necessarily non-empty) sets $(R_L,\leq_L)$ , the lower row, and $(R_U,\leq_U)$ , the upper row of $p$ , a set-theoretical partition $\pi$ of a disjoint union $R_L\sqcup R_U$ of $R_L$ and $R_U$ and a mapping $c:R_L\sqcup R_U\to \{\circ,\bullet\}$ , the coloring of $p$ , such that $p=(\pi,c)$ .

The set of all two-colored partitions is denoted by $\Pscr^{\circ\bullet}$ .

The elements of $R_L$ are called lower points and those of $R_U$ upper points. And for all $k,\ell\in \{0\}\cup \N$ we write $\Pscr^{\circ,\bullet}(k,\ell)$ for the set of all two-colored partitions with $k$ upper and $\ell$ lower points.

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Two-Colored Partition

Definition

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