CombinatorialCase002
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Jump to navigationJump to searchThis 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 ⎕IOsensitive
 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 nonnegative 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⌈MN) N