If $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ 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 $\lambda = (1^{m_1},2^{m_2},3^{m_3},\ldots)$ .

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