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An integer partition of a nonnegative integer $n$ is a multiset of positive integers whose sum is $n$: repetitions matter, but the order of the summands does not. Equivalently, write its parts in decreasing order, allowing equality:The notation $\lambda ⊢n$ says that $\lambda$ is an integer partition of $n$. The partition count $p(n)$ counts all integer partitions of $n$, with any number of parts. Zero has one empty integer partition, so $p(0)=1$. The partition of an integer article develops the same objects in representation theory of the symmetric group.^[ref]

Computed example. The seven integer partitions of five are exactlyThus $p(5)=7$. Sorting each list of positive summands into decreasing order prevents repetitions in this enumeration. Grouping by largest part verifies completeness: largest parts $5,4,3,2,1$ contribute respectively $1,1,2,2,1$ possibilities. For largest part $3$, the remaining $2$ is either $2$ or $1+1$; for largest part $2$, the remaining $3$ uses only $2$ and $1$, giving $2+1$ or $1+1+1$.

An integer partition keeps sizes without named positions. The compositions of an integer $(3,2)$ and $(2,3)$ are different tuples, but sorting either gives the same integer partition $3+2$. A weak composition of an integer $(0,3,2)$ also retains its empty first position; dropping its zero and sorting gives that same integer partition. Conversely, a set partition records which distinct elements belong to each block. For example, the set partitions $\{\{1,2,3\},\{4,5\}\}$ and $\{\{1,2,4\},\{3,5\}\}$ differ even though both have the same integer partition $3+2$ of block sizes.

The Ferrers diagram makes an integer partition into a shape. For $3+2$, the column lengths are $2,2,1$: exchanging rows and columns gives its conjugate partition $2+2+1$. Exchanging again recovers $3+2$, so this operation is a bijection. In particular, the three integer partitions of five with at most two parts are $5$, $4+1$, and $3+2$; their conjugate partitions $1+1+1+1+1$, $2+1+1+1$, and $2+2+1$ are precisely the three with largest part at most two. This visible symmetry explains the equal counts for these two restrictions. The square-cell version is the existing Young diagram.

Counting by a generating function. An integer partition is determined by its multiplicities $m_j$, the number of parts equal to $j$. Its size condition is ${\sum }_{j\ge 1}j{m}_j=n$. Consequently its ordinary generating function isProof outline. Choose the term $q^{j{m}_j}$ from the $j$th geometric series. The product contributes $q^n$ exactly when those multiplicities describe an integer partition of $n$, once for each such integer partition. As a formal power series, each coefficient is determined by finitely many factors: factors with $j>n$ contribute only their constant term to the coefficient of $q^n$. For $n=5$, multiplication through $j=5$ gives $1+q+2q^2+3q^3+5q^4+7q^5$ up to degree five, independently recovering $p(5)=7$. Restricting the factors gives the generating function for partitions with bounded parts. This is useful because restrictions on summands become choices of factors, while conjugate partitions turn restrictions on numbers of rows into restrictions on row lengths.^[ref]

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 sequenceWe write $\lambda ⊢n$, 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 areThis 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 ^[ref] documents this early development.

This representation is a distinguished example in the representation theory of the symmetric group.

The permutation representation of $S_n$ on ${\mathbb{C}}^n$ decomposes asThe first summand is the trivial representation and the second is the $(n-1)$-dimensional standard representation.

A Young tableau is a filling of a Young diagram with entries, typically positive integers. The shape is the partition of an integer giving its row lengths. We draw Young tableaux in the English convention of Young diagram, with left-aligned rows arranged from top to bottom. A filling need not obey increasing conditions unless these are specified: standard Young tableaux use each of $1,\dots ,n$ exactly once and strictly increase along rows and down columns; Semistandard Young tableaux permit repetitions and require weakly increasing rows and strictly increasing columns.

In the constructions for a symmetric group involving Young symmetrizers, tabloids, and polytabloids, an unqualified numbered Young tableau means a bijective filling by $1,\dots ,n$, with no increasing condition imposed. This narrower convention is compatible with the broader use needed for Semistandard Young tableaux. Historically, Young's work connected these fillings to representation theory of the symmetric group; in particular, the equality between standard Young tableau counts and irreducible representation degrees over the complex numbers predates the modern hook-length formula. See Stanley's Algebraic Combinatorics, historical notes to Chapter 8.