conjpart - Maple Help

combinat

 conjpart
 conjugate partition

 Calling Sequence conjpart(p)

Parameters

 p - partition; non-decreasing list of positive integers

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{combinat}\right):$
 > $\mathrm{conjpart}\left(\left[2,3,5\right]\right)$
 $\left[{1}{,}{1}{,}{2}{,}{3}{,}{3}\right]$ (1)
 > $\mathrm{conjpart}\left(\left[1,1,2,3,3\right]\right)$
 $\left[{2}{,}{3}{,}{5}\right]$ (2)
 > $\mathrm{conjpart}\left(\left[1,2,3\right]\right)$
 $\left[{1}{,}{2}{,}{3}\right]$ (3)