part of a partition

If $\lambda =({\lambda }_1,{\lambda }_2,\mathrm{\dots },{\lambda }_k)$ is an integer partition, then each ${\lambda }_j$ is a part of $\lambda$. The length of $\lambda$ is defined as the number of its parts.If $m_j$ is the number of parts equal to $j$, then the partition $\lambda$ is also written as $\lambda =(1^{m_1},2^{m_2},3^{m_3},\mathrm{\dots })$.

For example, if $\lambda =(5,4,4,4,3,3,3,3,3,1,1)$ then we also write $\lambda =(1^2,3^5,4^3,5^1)$.