
Calling Sequence


PartitionPartCount(n, opts)


Parameters


n



posint; integer to partition

opts



(optional) equation(s) of the form option = value; specify options for the PartitionPartCount command





Options



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 partcount form, in reverse lexicographic order.

•

A partition of integer n in partcount 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.
>

$P\u2254\mathrm{PartitionPartCount}\left(n\right)\:$

>

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

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.
>

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

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

$\mathrm{seq}\left(\mathrm{add}\left(kp\left[k\right]\,k=1..n\right)\,p=P\right)$

${8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}{,}{8}$
 (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.



