CombinatorialCase002: Difference between revisions
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(Created page with "This case produces the Partitions of the number L into exactly R parts. * <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2...") |
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==Identities== | ==Identities== | ||
Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts (<apll>001</apll>) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts (<apll>002</apll>), these cases are related by the following identity (after sorting the rows): | Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts ([[CombinatorialCase001|<apll>001</apll>]]) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts ([[CombinatorialCase002|<apll>002</apll>]]), these cases are related by the following identity (after sorting the rows): | ||
<apll>002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll> | <apll>002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll> | ||
Revision as of 20:56, 29 April 2017
This case produces the Partitions of the number L into exactly R parts.
- L unlabeled balls (0), R 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 L PN R where L PN R calculates the number of Partitions of the number L into exactly R 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 L into R parts
⍝ Unlabeled balls & boxes, ≥1 #bpb
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 L into R non-negative parts (001) is the same as partitions of L+R into R positive parts (002), these cases are related by the following identity (after sorting the rows):
002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R
