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partition

A partition is any (finite or infinite) sequence of nonnegative integers in decreasing order with finitely many nonzero terms
$\lambda =({\lambda }_1,{\lambda }_2,\dots )$ where ${\lambda }_1\ge {\lambda }_2\ge \dots$
The trailing zeros do not affect when two partitions are equal (so we can express any partition as finitely many positive integers).

We sometimes abbreviate as

This means that there are $m_r$ (multiplicity of a partition) copies of $r$.

The parts of $\lambda$ refer to each nonzero ${\lambda }_i$.
The length of $\lambda$ is the number of parts.
The weight of $\lambda$, $|\lambda |$, is the sum of the parts.
If the weight of $\lambda$ is $n$, then we say that $\lambda$ is a partition of $n$ and we denote this by $\lambda ⊢n$.
The set of all partitions of $n$ is denoted by ${\mathcal{P}}_n$.

Properties

• $\lambda$ partition if and only if $\lambda \ge w(\lambda )$ for all $w\in S_n$ permutation under the natural (dominance partial) ordering on partitions

See also

• n(lambda)

References