CombinatorialCase002: Difference between revisions
From NARS2000
Jump to navigationJump to search
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
This case produces the '''Partitions of the number | 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) | * <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2) | ||
Revision as of 20:43, 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:
|
|
|
|
|
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 # 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 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
