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Table of Contents
Partition of a set
Definition
Let 
be a finite set. A partition 
of is a decomposition of into disjoint non-empty subsets. That is
where 
, 
, and 
for any 
, 
. The subsets 
are called blocks of the partition .
The set of all partitions of is denoted 
. In combinatorics, we usually take the canonical choice 
, for which we denote 
. When using partitions to build diagramatical categories (such as the category of all partitions), we consider partitions of a set 
and denote 
.
Partition of a set should not be confused with a partition of a number. Any partition of a set induces a partition of a number if we remember only the sizes of the sets. Indeed, if 
is a partition of , then 
is a partition of 
.
Often we consider the set to be a totally ordered set. In this case, we can define the notion of a non-crossing partition. A partition of a totally ordered set is called non-crossing if for every four elements 
of having 
and 
implies 
. The name non-crossing is motivated by the graphical representation of such partitions (see below).
Representing partitions
