combinat
conjpart
conjugate partition
Calling Sequence
Parameters
Description
Examples
conjpart(p)
p

partition; nondecreasing list of positive integers
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 antidiagonal, 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.
$\mathrm{with}\left(\mathrm{combinat}\right)\:$
$\mathrm{conjpart}\left(\left[2\,3\,5\right]\right)$
$\left[{1}{\,}{1}{\,}{2}{\,}{3}{\,}{3}\right]$
$\mathrm{conjpart}\left(\left[1\,1\,2\,3\,3\right]\right)$
$\left[{2}{\,}{3}{\,}{5}\right]$
$\mathrm{conjpart}\left(\left[1\,2\,3\right]\right)$
$\left[{1}{\,}{2}{\,}{3}\right]$
See Also
combinat[encodepart]
combinat[inttovec]
combinat[numbpart]
combinat[partition]
combinat[randpart]
Definition/partition
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