This procedure counts the number of partitions of an integer n, that is, the number of ways n can be split into sums without regard to order.

See the partition function that constructs the partitions. Note that numbpart(n) = nops(partition(n)) for $0\le n$.

The command with(combinat,numbpart) allows the use of the abbreviated form of this command.

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

$\mathrm{numbpart}\left(5\right)$

$7$

$\mathrm{partition}\left(5\right)$

$\left[\left[1\,1\,1\,1\,1\right]\,\left[1\,1\,1\,2\right]\,\left[1\,2\,2\right]\,\left[1\,1\,3\right]\,\left[2\,3\right]\,\left[1\,4\right]\,\left[5\right]\right]$

$\mathrm{numbpart}\left(6\,3\right)$

$7$

$\mathrm{partition}\left(6\,3\right)$

$\left[\left[1\,1\,1\,1\,1\,1\right]\,\left[1\,1\,1\,1\,2\right]\,\left[1\,1\,2\,2\right]\,\left[2\,2\,2\right]\,\left[1\,1\,1\,3\right]\,\left[1\,2\,3\right]\,\left[3\,3\right]\right]$