CombinatorialCase002: Difference between revisions

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This case produces the '''Partitions of the number L into exactly R parts'''.
This case produces the '''Partitions of the number M into exactly N parts'''.


* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)
* <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>L PN R</apll> where <apll>L PN R</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll>L</apll> into exactly <apll>R</apll> parts.
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) 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 L into R parts
       ⍝ 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
Line 92: Line 92:
==Identities==
==Identities==


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):
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 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll>
<apll>002 1‼M N ↔ ⊃1+R↑¨001 1‼(0⌈M-N) N</apll>

Revision as of 16:17, 14 May 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:






 
 
 
 
 
 
 
 
 
 
     
 




 
 
 
 

 
 
 
 
 
     
 
 



 
 
 


 
 
 
 
 
     
 
 



 
 
 
 

 
 
 
 

     
 
 
 


 
 
 


 
 
 
 

     

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