compile = truefalse

True means compile the iterator. The default is true.

maxparts = posint

Specifies the maximum number of parts in each partition. Must be greater than one and less than or equal to n. The default is n.

parts = nonnegint

Specifies the number of parts in each partition. Must be either 0 (the default) or greater than one and less than or equal to n. The default, 0, is ignored. A non-zero value cannot be used with the maxparts option.

rank = nonnegint

Specify the starting rank of the iterator. The default is one. The starting rank reverts to one when the iterator is reset, reused, or copied.

The SetPartitions command returns an iterator that generates partitions of a set of integers from one to n. The output consists of the restricted growth strings of length n in lexicographic order.

A restricted growth string of length n corresponds to a partition of the set of integers 1 to n. Each integer in the output Array designates the set to which the index belongs; the sets are numbered starting at 0. For example, $⟨0\,0\,1\,2⟩$ corresponds to the partition $\left\{1\,2\right\},\left\{3\right\},\left\{4\right\}$.

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

$M≔\mathrm{SetPartitions}\left(4\right)\:$

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

 1: 0 0 0 0
 2: 0 0 0 1
 3: 0 0 1 0
 4: 0 0 1 1
 5: 0 0 1 2
 6: 0 1 0 0
 7: 0 1 0 1
 8: 0 1 0 2
 9: 0 1 1 0
10: 0 1 1 1
11: 0 1 1 2
12: 0 1 2 0
13: 0 1 2 1
14: 0 1 2 2
15: 0 1 2 3

$N≔\mathrm{Object}\left(M\,'\mathrm{maxparts}'=3\right)\:$

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

 1: 0 0 0 0
 2: 0 0 0 1
 3: 0 0 1 0
 4: 0 0 1 1
 5: 0 0 1 2
 6: 0 1 0 0
 7: 0 1 0 1
 8: 0 1 0 2
 9: 0 1 1 0
10: 0 1 1 1
11: 0 1 1 2
12: 0 1 2 0
13: 0 1 2 1
14: 0 1 2 2

$N≔\mathrm{Object}\left(M\,'\mathrm{parts}'=2\right)\:$

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

1: 0 0 0 1
2: 0 0 1 0
3: 0 0 1 1
4: 0 1 0 0
5: 0 1 0 1
6: 0 1 1 0
7: 0 1 1 1

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

$15$

Knuth, Donald Ervin. The Art of Computer Programming, volume 4, fascicle 3; generating all combinations and partitions, sec. 7.2.1.5, generating all set partitions, algorithm H, restricted growth strings in lexicographic order, pp. 62-63 and p. 124 ex. 1.

The Iterator[SetPartitions] 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.