The conjpart(p) command computes and returns the conjugate partition of p.

A partition $p=\[i_1,i_2,...,i_m\]$ of a positive integer $n$ may be represented visually by its Ferrer's diagram. This is a diagram composed of dots in rows, in which the $k$th row consists of $i_k$ dots, for $k=1..m$. The total number of dots in the diagram is equal to the number $n$. For example, the partition $\left[2\,3\,5\right]$ of $10$ has the Ferrer's diagram:

consisting of ten dots arranged in three rows, with two dots in the first row, three dots in the second, and five dots in the third row.

Two partitions (of a positive integer $n$) are said to be conjugates if their Ferrer's diagrams are conjugate, which means that one is obtained from the other, by reflection along the anti-diagonal, by writing the rows as columns and columns as rows. For example, the conjugate of the Ferror diagram above is:

which represents the partition $\left[1\,1\,2\,3\,3\right]$. Therefore, the partitions $\left[2\,3\,5\right]$ and $\left[1\,1\,2\,3\,3\right]$ are conjugate partitions.

$\mathrm{with}\left(\mathrm{combinat}\right)\:$

$\mathrm{conjpart}\left(\left[2\,3\,5\right]\right)$

$\left[1\,1\,2\,3\,3\right]$

$\mathrm{conjpart}\left(\left[1\,1\,2\,3\,3\right]\right)$

$\left[2\,3\,5\right]$

$\mathrm{conjpart}\left(\left[1\,2\,3\right]\right)$

$\left[1\,2\,3\right]$