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 $\left(a_1\&comma;\dots \&comma;a_k\right)$ such that $n\&equals;\sum _{j\&equals;1}^k{a}_j$ and $0<a_j\le n$ for $j\in \left\{1\&comma;\dots \&comma;k\right\}$.

The n parameter is the integer to partition.

The optional m parameter is the maximum integer that can appear in any partition. The default is n.

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.

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

$P≔\mathrm{Partition}\left(8\right)\&colon;$

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

 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

$P≔\mathrm{Partition}\left(8\&comma;5\right)\&colon;$

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

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

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

$18$

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

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

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.

The Iterator[Partition] command was updated in Maple 2022.

The n and m parameters were updated in Maple 2022.