CombinatorialCase002

This case produces the Partitions of the number M into exactly N parts.

• M unlabeled balls (0), N unlabeled boxes (0), at least one ball per box (2)
• Not ⎕IO-sensitive
• Counted result is an integer scalar
• Generated result is an integer matrix.

The count for this function is M PN N where M PN N calculates the number of Partitions of the number M into exactly N parts.

For example:

If we have 8 unlabeled balls (●●●●●●●●) and 3 unlabeled boxes with at least one ball per box, there are 5 (↔ 8 PN 3) ways to meet these criteria:

● ● ● ● ● ●           ●           ●

● ● ● ● ●         ● ●           ●

● ● ● ●       ● ● ●           ●

● ● ● ●         ● ●         ● ●

● ● ●       ● ● ●         ● ●

The diagram above corresponds to

      002 1‼8 3
6 1 1
5 2 1
4 3 1
4 2 2
3 3 2

      ⍝ Partitions of M into N parts
      ⍝ Unlabeled balls & boxes, ≥1 # Balls per Box
      002 1‼5 5
1 1 1 1 1
      002 1‼5 4
2 1 1 1
      002 1‼5 3
3 1 1
2 2 1
      002 1‼5 2
4 1
3 2
      002 1‼5 1
5

Identities

Because partitions of M into N non-negative parts (001) is the same as partitions of M+N into N positive parts (002), these cases are related by the following identity (after sorting the rows):

002 1‼M N ↔ ⊃1+R↑¨001 1‼(0⌈M-N) N