= Partition of an integer
{title2=$\lambda\vdash n$}
{wiki=Integer_partition}
= Partitions of an integer
{synonym}
A partition of a <nonnegative integer> $n$ records a way to write $n$ as a sum of positive <integers>, with the order of the summands ignored. Its canonical notation is a finite <sequence>
$$
\lambda=(\lambda_1,\ldots,\lambda_\ell),\qquad
\lambda_1\geq\cdots\geq\lambda_\ell>0,\qquad
|\lambda|:=\sum_{i=1}^{\ell}\lambda_i=n.
$$
We write $\lambda\vdash n$, call the $\lambda_i$ its parts, and call $\ell=\ell(\lambda)$ its length. Trailing zero parts may be appended for calculations without changing the <partition of an integer>; zero has the unique empty <partition of an integer>, of length zero. Thus $(3,2)$, $(3,2,0)$ and $5=2+3$ describe the same <partition of an integer> of five. The <Young diagram> below makes its two parts visible as rows of three and two cells.
The seven <partitions of an integer> of five are
$$
(5),\ (4,1),\ (3,2),\ (3,1,1),\ (2,2,1),\ (2,1,1,1),\ (1,1,1,1,1).
$$
This is a different object from a <set partition>: $(3,2)$ specifies block sizes, whereas a <set partition> must also specify which elements belong to each block.
Why this is useful: a <partition of an integer> is a compact description of a shape, yet it also labels an <irreducible representation> of a <symmetric group> over the <complex numbers>, realized by a <Specht module>. It therefore connects arithmetic decompositions, <Young diagrams>, and <representation theory of the symmetric group>. The <generating function for partitions with bounded parts> gives another way to count such shapes. Historically, Euler studied these counting problems using <generating functions>; his https://scholarlycommons.pacific.edu/euler-works/191/[De partitione numerorum, written in 1750 and published in 1753, with the original scan in the Euler Archive]{ref} documents this early development.
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