Conjugates of a partition $n$

Question

Is there any easiest way finding of conjugate of a partition $n$ (Except using Ferrers diagram)?

e.g I can find the conjugate of a partition $a=[4,4,1]$ using Ferrers diagram, and I obtain $a^*=[3,2,2,2]$

\begin{equation} \begin{split} &**** \ \ \ &***\\ &**** \to \ &*\,*\\ &* \ \ &*\,*\\ & &*\,* \end{split} \end{equation}

Answer 1

Given a partition $(\lambda_1,\lambda_2,\dots)$, then $$ (\lambda^*)_i=\# \{j:\lambda_j\ge i\} $$ In your example,

• There are $3$ entries of $[4,4,1]$ which are $1$ or more,
• There are $2$ entries of $[4,4,1]$ which are $2$ or more,
• There are $2$ entries of $[4,4,1]$ which are $3$ or more,
• There are $2$ entries of $[4,4,1]$ which are $4$ or more,

so $a^*=[3,2,2,2]$.