CombinatorialCase011

From NARS2000
Revision as of 16:36, 29 April 2017 by WikiSysop (talk | contribs) (Created page with "This case produces <apll>L</apll> multicombinations of <apll>R</apll> items. A multicombination is a collection of Multisets (sets which may contain repeated elements) ac...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

This case produces L multicombinations of R items. A multicombination is a collection of Multisets (sets which may contain repeated elements) according to certain criteria. In particular, it produces a matrix whose rows are multisets of length L, from the values ⍳R.

  • L unlabeled balls (0), R labeled boxes (1), any # balls per box (1)
  • Sensitive to ⎕IO
  • Counted result is an integer scalar
  • Generated result is an integer matrix.

The count for this function is L!L+R-1.

For example:

If we have 2 unlabeled balls (●●) and 3 labeled boxes (123) with any # of balls per box, there are 6 (↔ 2!2+3-1) ways to meet these criteria:


       
1 2 3
 
 
   
1 2 3
 
     
1 2 3
   
   
1 2 3
     
 
1 2 3
       
1 2 3

The diagram above corresponds to:

      011 1‼2 3
1 1
1 2
1 3
2 2
2 3
3 3
      ⍝ L Multicombinations of R items
      ⍝ Unlabeled balls, labeled boxes, any #bpb
      011 1‼3 3
1 1 1
1 1 2
1 1 3
1 2 2
1 2 3
1 3 3
2 2 2
2 2 3
2 3 3
3 3 3
      011 1‼3 2
1 1 1
1 1 2
1 2 2
2 2 2
      011 1‼3 1
1 1 1

In general, this case is related to that of Combinations (010) via the following identities:

010 1‼L R ↔ (011 1‼L,R-L-1)+[⎕IO+1] 0..L-1
011 1‼L R ↔ (010 1‼L,L+R-1)-[⎕IO+1] 0..L-1