PartitionPartCount - Maple Help

Iterator

 PartitionPartCount
 generate partitions of an integer in part-count form

 Calling Sequence PartitionPartCount(n, opts)

Parameters

 n - nonnegint; integer to partition opts - (optional) equation(s) of the form option = value; specify options for the PartitionPartCount command

Options

 • compile = truefalse
 True means compile the iterator. The default is true.

Description

 • The PartitionPartCount command returns an iterator that generates all partitions of the integer n in part-count form, in reverse lexicographic order.
 • A partition of integer n in part-count form is a sequence of integers $\left({c}_{1},\dots ,{c}_{n}\right)$ such that $n=\sum _{k=1}^{n}k{c}_{k}$ and $0\le {c}_{k}\le n$ for $k\in \left\{1,\dots ,n\right\}$.
 • The n parameter is the integer to partition.
 • The output of the iterator is an array of fixed length n.

Methods

In addition to the common iterator methods, this iterator object has the following methods. The self parameter is the iterator object.

 • Number(self): return the number of iterations required to step through the iterator, assuming it started at rank one.

Examples

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

Iterate through the partitions of 8.

 > $n≔8:$
 > $P≔\mathrm{PartitionPartCount}\left(n\right):$
 > $\mathrm{Print}\left(P,'\mathrm{showrank}'\right):$
 1: 8 0 0 0 0 0 0 0  2: 6 1 0 0 0 0 0 0  3: 4 2 0 0 0 0 0 0  4: 2 3 0 0 0 0 0 0  5: 0 4 0 0 0 0 0 0  6: 5 0 1 0 0 0 0 0  7: 3 1 1 0 0 0 0 0  8: 1 2 1 0 0 0 0 0  9: 2 0 2 0 0 0 0 0 10: 0 1 2 0 0 0 0 0 11: 4 0 0 1 0 0 0 0 12: 2 1 0 1 0 0 0 0 13: 0 2 0 1 0 0 0 0 14: 1 0 1 1 0 0 0 0 15: 0 0 0 2 0 0 0 0 16: 3 0 0 0 1 0 0 0 17: 1 1 0 0 1 0 0 0 18: 0 0 1 0 1 0 0 0 19: 2 0 0 0 0 1 0 0 20: 0 1 0 0 0 1 0 0 21: 1 0 0 0 0 0 1 0 22: 0 0 0 0 0 0 0 1

Compute the number of iterations.

 > $\mathrm{Number}\left(P\right)$
 ${22}$ (1)

Add the elements of each partition to verify they sum to n.

 > $\mathrm{seq}\left(\mathrm{add}\left(k{p}_{k},k=1..n\right),p=P\right)$
 ${8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}$ (2)

References

 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 C, p. 110, ex. 5.

Compatibility

 • The Iterator[PartitionPartCount] command was introduced in Maple 2016.