The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group.

A partition of ${\displaystyle 2k}$ elements labelled ${\displaystyle 1,{\bar {1}},2,{\bar {2}},\dots ,k,{\bar {k}}}$ is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset ${\displaystyle \{{\bar {1}},{\bar {4}},{\bar {5}},6\}}$ gives rise to the lines ${\displaystyle {\bar {1}}-{\bar {4}},{\bar {4}}-{\bar {5}},{\bar {5}}-6}$, and could equivalently be represented by the lines ${\displaystyle {\bar {1}}-6,{\bar {4}}-6,{\bar {5}}-6,{\bar {1}}-{\bar {5}}}$ (for instance).

For ${\displaystyle n\in \mathbb {C} }$ and ${\displaystyle k\in \mathbb {N} ^{*}}$, the partition algebra ${\displaystyle P_{k}(n)}$ is defined by a ${\displaystyle \mathbb {C} }$-basis made of partitions, and a multiplication given by diagram concatenation. The concatenated diagram comes with a factor ${\displaystyle n^{D}}$, where ${\displaystyle D}$ is the number of connected components that are disconnected from the top and bottom elements.

The partition algebra ${\displaystyle P_{k}(n)}$ is generated by ${\displaystyle 3k-2}$ elements of the type

These generators obey relations that include

Other elements that are useful for generating subalgebras include

In terms of the original generators, these elements are

The partition algebra ${\displaystyle P_{k}(n)}$ is an associative algebra. It has a multiplicative identity