Definition & Example

An integer partition $\lambda$ of $n \in \mathbb{N}_+$ is a sequence $\lambda = (\lambda_1,\ldots,\lambda_k)$ such that $\lambda_i \in \mathbb{N}_{+}$, $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_k$ and $\sum_{1 \leq i \leq k} \lambda_i = n$.
We write $\lambda \vdash n$ if $\lambda$ is a partition of $n$, and denote the set of integer partitions of $n$ by $\mathbf{P}_n$.

the 7 Integer partitions of size 5

`[5]`	`[4,1]`	`[3,2]`	`[3,1,1]`	`[2,2,1]`	`[2,1,1,1]`	`[1,1,1,1,1]`

Integer partitions are graphically represented by their Ferrers diagram (or Young diagram) as a collection of boxes.
The number of integer partitions $p(n) = |\mathbf{P}_n|$ is A000041, and the number of integer partitions into distinct parts is A000009. Its generating function is
$$\displaystyle \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty \frac{1}{1-x^k}.$$
This is also known as the Euler's generating function which allows $p(n)$ to be calculated for any $n$, see [Wil00].

Properties

Let $p_n(r)$ be the number of partitions of $n$ in which the largest part is $r$. Let $q_n(r)$ be the number of partitions of $n-r$ which no part is greater than r. Then

$$p_n(r)=q_n(r)$$

Let $p_n^s$ denote the number of self conjugate partitions of n and let $p_n^t$ be the number of partitions of n into distinct odd parts. Then

$$p_n^s=p_n^t$$

Let $p_n^o$ be the number of partitions of n into odd parts, and let $p_n^d$ be the number of partitions of n into distinct parts. Then

$$p_n^o=p_n^d,$$

see [Bru10].

The elements of $\mathbf{P}_n$ index the number of irreducible representations of the symmetric group $\mathbf{S}_n$ over a field of characteristic $0$.
The number of non-isomorphic abelian groups of order $n=\sum_ip_i^{j_i}$ where $p_i$ are distinct primes and $j_i$ are positive integers is given by $\prod_ip(j_i)$ A000688
Congruence partitions are partitions that are similar under modulation. For example:
$$p(5k+4)= 0 (mod\text{ }5)$$

$$p(7k+5)= 0 (mod\text{ }7)$$

$$p(11k+6)=0 (mod\text{ }11)$$
are just a couple examples proven by Srinivasa Ramanujan.
The approximation formula for p(n):
$$\displaystyle p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right) \mbox { as } n\rightarrow \infty$$
For many combinatorial statistics and bijections see [Pak02]

[Bru10] R. Brualdi, Partition numbers, Combinatorics Fifth Edition (2010)
[Pak02] I. Pak, Partition bijections, a survey, Ramanujan J. 12(1) (2006) p. 5-75
[Wil00] S. Wilf, Basic generating functions, Lectures on Integer Partitions (2000)

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