CombinatorialCase001

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

• M unlabeled balls (0), N unlabeled boxes (0), any # of balls per box (1)
• Not ⎕IO-sensitive
• Counted result is an integer scalar
• Generated result is a nested vector of integer vectors.

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

For example:

If we have 6 unlabeled balls (●●●●●●) and 3 unlabeled boxes with any # of balls per box, there are 7 (↔ (6+3)PN 3) ways to meet these criteria:

● ● ● ● ● ●

● ● ● ● ●           ●

● ● ● ●         ● ●

● ● ● ●           ●           ●

● ● ●       ● ● ●

● ● ●         ● ●           ●

● ●         ● ●         ● ●

The diagram above corresponds to the nested array

      ⍪001 1‼6 3
 6
 5 1
 4 2
 4 1 1
 3 3
 3 2 1
 2 2 2
      ⍝ Partitions of M into at most N parts
      ⍝ Unlabeled balls & boxes, any # Balls per Box
      ⍪001 1‼5 5
 5
 4 1
 3 2
 3 1 1
 2 2 1
 2 1 1 1
 1 1 1 1 1
      ⍪001 1‼5 4
 5
 4 1
 3 2
 3 1 1
 2 2 1
 2 1 1 1
      ⍪001 1‼5 3
 5
 4 1
 3 2
 3 1 1
 2 2 1
      ⍪001 1‼5 2
 5
 4 1
 3 2
      ⍪001 1‼5 1
 5

Identities

As shown in Wikipedia, (M+N)PN N ↔ +/M PN¨0..N.

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:

001 1‼M N ↔ (⊂[⎕IO+1] ¯1+002 1‼(M+N) N)~¨0