Numeric partition

A partition of a number $𝑛 \in ℕ 0$ is just a set of numbers adding to $𝑛$ . #m/def/num In order to have uniqueness, a partition is defined as an infinite sequence $(𝜆 𝑖) \infty 𝑖 = 1$ in $ℕ 0$ that adds to $𝑛$ and is non-increasing.

$𝜆 𝑖 \geq 𝜆 𝑖 + 1$ for all $𝑖 \in ℕ$
$\sum 𝑛 𝑖 = 1 𝜆 𝑖 = 𝑛$

A graphical portrayal of a partition is a Young diagram.

Ordering

Partitions can be ordered by

$𝜆 = 𝜇$ iff $𝜆 𝑖 = 𝜇 𝑖$ for all $𝑖 \in ℕ$
$𝜆 > 𝜇$ iff the first non-vanishing $𝜆 𝑖 - 𝜇 𝑖$ is positive

#state/tidy | #lang/en | #SemBr