✎ two-colored_partition [CMQG Wiki]

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====== 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 group|compact matrix quantum groups]], the so-called [[unitary_easy_quantum_group|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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