compile = truefalse

True means compile the iterator. The default is true.

The Partition command returns an iterator that generates all partitions of the integer n, in reverse lexicographic order.

A partition of integer n is a sequence of integers $\left(a_1\&comma;\dots \&comma;a_m\right)$ such that $n\&equals;\sum _{k\&equals;1}^m{a}_k$ and $0<a_k\le n$ for $k\in \left\{1\&comma;\dots \&comma;m\right\}$.

The n parameter is the integer to partition.

The output of the iterator is an array of fixed length n. The partition is in the indices 1 to $\mathrm{length}\left(P\right)$, where P is the assigned iterator.

$\mathrm{with}\left(\mathrm{Iterator}\right)\&colon;$

$n≔8\&colon;$

$P≔\mathrm{Partition}\left(n\right)\&colon;$

$\mathrm{Print}\left(P\&comma;'\mathrm{showrank}'\right)\&colon;$

 1: 8
 2: 7 1
 3: 6 2
 4: 6 1 1
 5: 5 3
 6: 5 2 1
 7: 5 1 1 1
 8: 4 4
 9: 4 3 1
10: 4 2 2
11: 4 2 1 1
12: 4 1 1 1 1
13: 3 3 2
14: 3 3 1 1
15: 3 2 2 1
16: 3 2 1 1 1
17: 3 1 1 1 1 1
18: 2 2 2 2
19: 2 2 2 1 1
20: 2 2 1 1 1 1
21: 2 1 1 1 1 1 1
22: 1 1 1 1 1 1 1 1

$\mathrm{Number}\left(P\right)$

$22$

$\mathrm{seq}\left(p\left[1..\mathrm{length}\left(M\right)\right]\&comma;p=P\right)$

$\left[\begin{array}{c}8\end{array}\right],\left[\begin{array}{c}7\end{array}\right],\left[\begin{array}{c}6\end{array}\right],\left[\begin{array}{c}6\end{array}\right],\left[\begin{array}{c}5\end{array}\right],\left[\begin{array}{c}5\end{array}\right],\left[\begin{array}{c}5\end{array}\right],\left[\begin{array}{c}4\end{array}\right],\left[\begin{array}{c}4\end{array}\right],\left[\begin{array}{c}4\end{array}\right],\left[\begin{array}{c}4\end{array}\right],\left[\begin{array}{c}4\end{array}\right],\left[\begin{array}{c}3\end{array}\right],\left[\begin{array}{c}3\end{array}\right],\left[\begin{array}{c}3\end{array}\right],\left[\begin{array}{c}3\end{array}\right],\left[\begin{array}{c}3\end{array}\right],\left[\begin{array}{c}2\end{array}\right],\left[\begin{array}{c}2\end{array}\right],\left[\begin{array}{c}2\end{array}\right],\left[\begin{array}{c}2\end{array}\right],\left[\begin{array}{c}1\end{array}\right]$

$\mathrm{seq}\left(\mathrm{add}\left(p\left[k\right]\&comma;k=1..\mathrm{length}\left(P\right)\right)\&comma;p=P\right)$

$8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8$

Knuth, Donald Ervin. The Art of Computer Programming, volume 4, fascicle 3; generating all combinations and partitions, sec. 7.2.1.4, generating all partitions, algorithm P, partitions in reverse lexicographic order.  The algorithm was corrected; Knuth_errata, p. 38.

The Iterator[Partition] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.