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--- category_of_all_partitions [2019/03/03 14:34]
d.gromada
+++ category_of_all_partitions [2021/11/23 11:56] (current)
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 ==== Partitions ====
-Let $k,l\in\mathbb{N}_0$, by a **partition** of $k$ upper and $l$ lower points we mean a partition of the set $\{1,\dots,k\}\sqcup\{1,\dots,l\}\approx\{1,\dots,k+l\}$, that is, a decomposition of the set of $k+l$ points into non-empty disjoint subsets, called **blocks**. The first $k$ points are called **upper** and the last $l$ points are called **lower**. The set of all partitions on $k$ upper and $l$ lower points is denoted $\Pscr(k,l)$. We denote the union $\Pscr:=\bigcup_{k,l\in\N_0}\Pscr(k,l)$.
+Let $k,l\in\mathbb{N}_0$, by a [[partition|partition]] of $k$ upper and $l$ lower points we mean herre a partition of the set $\{1,\dots,k\}\sqcup\{1,\dots,l\}\approx\{1,\dots,k+l\}$, that is, a decomposition of the set of $k+l$ points into non-empty disjoint subsets, called **blocks**. The first $k$ points are called **upper** and the last $l$ points are called **lower**. The set of all partitions on $k$ upper and $l$ lower points is denoted $\Pscr(k,l)$. We denote the union $\Pscr:=\bigcup_{k,l\in\N_0}\Pscr(k,l)$.
 We illustrate partitions graphically by putting $k$ points in one row and $l$ points on another row below and connecting by lines those points that are grouped in one block. All lines are drawn between those two rows.