CombinatorialCase012

This case produces Compositions of the number M into N parts. A composition is a way of representing a number as the sum of all positive integers, in this case it’s a way of representing M as the sum of N positive integers. It can also be thought of as a partition of M into N ordered parts.

• M unlabeled balls (0), N labeled boxes (1), 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-N)!M-1.

For example:

If we have 5 unlabeled balls (●●●●●) and 3 labeled boxes (123) with at least one ball per box, there are 6 (↔ (5-3)!5-1) ways to meet these criteria:

●     ● ● ● ● 1 2 3

●   ● ●   ● ● 1 2 3

● ● ● ●     ● 1 2 3

● ●     ●   ● ● 1 2 3

● ●   ● ●     ● 1 2 3

● ● ●     ●     ● 1 2 3

The diagram above corresponds to

      012 1‼5 3
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
      ⍝ Compositions of M into N parts
      ⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box
      012 1‼5 5
1 1 1 1 1
      012 1‼5 4
1 1 1 2
1 1 2 1
1 2 1 1
2 1 1 1
      012 1‼5 3
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
      012 1‼5 2
1 4
2 3
3 2
4 1
      012 1‼5 1
5

In general, because the counts of both compositions (012) and combinations (010) is a binomial coefficient, there might be a mapping between the two, and indeed there is, as seen by the following identities:

010 1‼M N ↔ +\0 ¯1↓012 1‼⍠1 N M+1
012 1‼M N ↔ ¯2-\(010 1‼⍠1 N M-1),M

where ‼⍠1 uses the Variant operator ⍠ to evaluate ‼ in origin 1.