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 ⎕IOsensitive
 Allows Lexicographic order
 Counted result is an integer scalar
 Generated result is an integer matrix.
The count for this function is (MN)!M1.
For example:
If we have 5 unlabeled balls (●●●●●) and 3 labeled boxes (123) with at least one ball per box, there are 6 (↔ (53)!51) ways to meet these criteria:






The diagram above corresponds to
12 1‼5 3 ⍝ Compositions in unspecified order 1 1 3 2 1 2 1 2 2 3 1 1 2 2 1 1 3 1 12 2‼5 3 ⍝ Compositions in Lexicographic order 1 1 3 1 2 2 1 3 1 2 1 2 2 2 1 3 1 1 12 3‼5 3 ⍝ Gray Code order for Compositions not implemented as yet NONCE ERROR 12 3‼5 3 ∧ ⍝ Compositions of M into N parts ⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box 12 2‼5 5 1 1 1 1 1 12 2‼5 4 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1 12 2‼5 3 1 1 3 1 2 2 1 3 1 2 1 2 2 2 1 3 1 1 12 2‼5 2 1 4 2 3 3 2 4 1 12 2‼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 M1),M
where ‼⍠1 uses the Variant operator ⍠ to evaluate ‼ in origin 1.