Partitions

Definition:The vector $\la=(\la_1,\dots,\la_n)$ is called a partition of $k$ if $\la_1\geq\dots\geq\la_n\geq 0$ and $\vert\la\vert=\la_1+\dots\la_n=k$. The number $k$ is called length, numbers $\la_i$ are called parts of $\la$.

Partitions can be represented by pictures called Young diagrams (or Ferrers diagrams). The Young diagram of $\la$ consists of $n$ rows of boxes aligned on the left, such that $i$-th row is right on $i+1$-st row. The length of $i$-th row is $\la_i$. The conjugate partition $\la'$ is the partition with the Young diagrams consisting of columns of lengths $\la_i$. For example $\la_1'$ is the number of nonzero parts of $\la$. If $\la=(3,3,1)$ then $\la'=(3,2,2)$. Also $\la''=\la$.

Exercise:Show that the number of partitions of $n$ with odd distinct parts equals to number of self conjugated partitions of $n$ (that is partitions $\la$ with the property $\la=\la'$). $\;\Box$

Definition:A partition $\la$ is said to be larger than a partition $\mu$ if $\vert\la\vert=\vert\mu\vert$ and we have

$$\la_1 \geq \mu_1$$

$$\la_1+\la_2 \geq \mu_1+\mu_2$$

$$\la_1+\la_2+\la_3 \geq \mu_1+\mu_2+\mu_3$$

$$\dots$$

The largest partition of length $k$ is $(k,0,0,\dots,0)$. If $k\leq n$ then the smallest partition of length $k$ is $(1,1,\dots,1,0,\dots,0)$.

Exercise:Show that $\la\geq\mu$ if and only if the Young diagrams of $\la$ can be obtained from Young diagram of $\mu$ by raising some boxes from lower rows to higher ones. $\;\Box$

Exercise:Find an example of two partitions of $6$, none of which is greater then another. $\;\Box$