CombinatorialCase002: Difference between revisions
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This case produces the '''Partitions of the number''' | This case produces the '''Partitions of the number M into exactly N parts'''. | ||
* <apll> | * <apll>M</apll> unlabeled balls (0), <apll>N</apll> unlabeled boxes (0), at least one ball per box (2) | ||
* Not <apll>⎕IO</apll>-sensitive | * Not <apll>⎕IO</apll>-sensitive | ||
* Counted result is an integer scalar | * Counted result is an integer scalar | ||
* Generated result is an integer matrix. | * Generated result is an integer matrix. | ||
The count for this function is <apll> | The count for this function is <apll>M PN N</apll> where <apll>M PN N</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_part_size_or_number_of_parts Partitions] of the number <apll>M</apll> into exactly <apll>N</apll> parts. | ||
For example: | For example: | ||
| Line 75: | Line 75: | ||
3 3 2 | 3 3 2 | ||
⍝ Partitions of | ⍝ Partitions of M into N parts | ||
⍝ Unlabeled balls & boxes, ≥1 # Balls per Box | ⍝ Unlabeled balls & boxes, ≥1 # Balls per Box | ||
002 1‼5 5 | 002 1‼5 5 | ||
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==Identities== | ==Identities== | ||
Because partitions of <apll> | Because partitions of <apll>M</apll> into <apll>N</apll> non-negative parts ([[CombinatorialCase001|<apll>001</apll>]]) is the same as partitions of <apll>M+N</apll> into <apll>N</apll> positive parts ([[CombinatorialCase002|<apll>002</apll>]]), these cases are related by the following identity (after sorting the rows): | ||
<apll>002 | <apll>002 1‼M N ↔ ⊃1+R↑¨001 1‼(0⌈M-N) N</apll> | ||
Latest revision as of 18:50, 19 June 2017
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:
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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
