maxparts = nonnegint

Specifies the maximum number of parts in each partition. Must be either 0 or greater than one. The default, 0, is ignored. When maxparts is used, some of the elements in the output may be zero.

parts = nonnegint

Specifies the number of parts in each partition. This option overrides the positional argument m; the default is m.

compile = truefalse

True means compile the iterator. The default is true.

The PartitionFixedSize command returns an iterator that generates all integer m-tuples, $a_1,\dots ,a_m$ such that $a_k\ge a_{k\+1}$, $a_m\ge 1$, and $\sum _{k=1}^m{a}_k=n$.

The partitions are visited in colex order, that is, the reflected sequence $a_m,\dots ,a_1$ is in lexicographic order.

The n parameter specifies the integer to partition.

The m parameter specifies the size of the partition. The option parts may also be used to specify the number of parts.

The cost to run completely through this iterator is at most $\mathrm{O}\left(3N\left(n\,m\right)+m\right)$, where $N\left(n\,m\right)$ is the number of partitions of $n$ of size $m$, cf. [1], theorem H, p. 50.

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

$M≔\mathrm{PartitionFixedSize}\left(12\,'\mathrm{parts}'=5\right)\:$

$\mathrm{Print}\left(M\,'\mathrm{showrank}'\right)\:$

 1: 8 1 1 1 1
 2: 7 2 1 1 1
 3: 6 3 1 1 1
 4: 5 4 1 1 1
 5: 6 2 2 1 1
 6: 5 3 2 1 1
 7: 4 4 2 1 1
 8: 4 3 3 1 1
 9: 5 2 2 2 1
10: 4 3 2 2 1
11: 3 3 3 2 1
12: 4 2 2 2 2
13: 3 3 2 2 2

$N≔\mathrm{Object}\left(M\,'\mathrm{maxparts}'=5\right)\:$

$\mathrm{Print}\left(N\,'\mathrm{showrank}'\right)\:$

 1: 12 0 0 0 0
 2: 11 1 0 0 0
 3: 10 2 0 0 0
 4: 9 3 0 0 0
 5: 8 4 0 0 0
 6: 7 5 0 0 0
 7: 6 6 0 0 0
 8: 10 1 1 0 0
 9: 9 2 1 0 0
10: 8 3 1 0 0
11: 7 4 1 0 0
12: 6 5 1 0 0
13: 8 2 2 0 0
14: 7 3 2 0 0
15: 6 4 2 0 0
16: 5 5 2 0 0
17: 6 3 3 0 0
18: 5 4 3 0 0
19: 4 4 4 0 0
20: 9 1 1 1 0
21: 8 2 1 1 0
22: 7 3 1 1 0
23: 6 4 1 1 0
24: 5 5 1 1 0
25: 7 2 2 1 0
26: 6 3 2 1 0
27: 5 4 2 1 0
28: 5 3 3 1 0
29: 4 4 3 1 0
30: 6 2 2 2 0
31: 5 3 2 2 0
32: 4 4 2 2 0
33: 4 3 3 2 0
34: 3 3 3 3 0
35: 8 1 1 1 1
36: 7 2 1 1 1
37: 6 3 1 1 1
38: 5 4 1 1 1
39: 6 2 2 1 1
40: 5 3 2 1 1
41: 4 4 2 1 1
42: 4 3 3 1 1
43: 5 2 2 2 1
44: 4 3 2 2 1
45: 3 3 3 2 1
46: 4 2 2 2 2
47: 3 3 2 2 2

1. 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 H, partitions into m parts, p. 38, and p. 110, answer 2.

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

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

The maxparts and parts options were introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.

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

The n and m parameters were updated in Maple 2022.