View source for CombinatorialCase012

This case produces '''Compositions of the number <apll>M</apll> into <apll>N</apll> parts'''.  A composition is a way of representing a number as the sum of all positive integers, in this case it’s a way of representing <apll>M</apll> as the sum of <apll>N</apll> positive integers.  It can also be thought of as a [https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_part_size_or_number_of_parts Partition] of <apll>M</apll> into <apll>N</apll> Ordered Parts.

* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at least one ball per box (2)
* Not <apll>⎕IO</apll>-sensitive
* Allows Lexicographic order
* Counted result is an integer scalar
* Generated result is an integer matrix.

The count for this function is <apll>(M-N)!M-1</apll>.

For example:

If we have 5 unlabeled balls (<span style="font-size: 2em;">●●●●●</span>) and 3 labeled boxes (<apll>123</apll>) with at least one ball per box, there are <apll>6</apll> (<apll>↔ (5-3)!5-1</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span>
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|}

The diagram above corresponds to

<pre>
      12 1‼5 3 ⍝ Compositions in unspecified order
1 1 3
2 1 2
1 2 2
3 1 1
2 2 1
1 3 1
      12 2‼5 3 ⍝ Compositions in Lexicographic order
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
      12 3‼5 3 ⍝ Gray Code order for Compositions not implemented as yet
NONCE ERROR
      12 3‼5 3
          ∧

      ⍝ Compositions of M into N parts
      ⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box
      12 2‼5 5
1 1 1 1 1
      12 2‼5 4
1 1 1 2
1 1 2 1
1 2 1 1
2 1 1 1
      12 2‼5 3
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
      12 2‼5 2
1 4
2 3
3 2
4 1
      12 2‼5 1
5
</pre>

In general, because the counts of both compositions ([[CombinatorialCase012|<apll>012</apll>]]) and combinations ([[CombinatorialCase010|<apll>010</apll>]]) is a binomial coefficient, there might be a mapping between the two, and indeed there is, as seen by the following identities:

<pre>
010 1‼M N ↔ +\0 ¯1↓012 1‼⍠1 N M+1
012 1‼M N ↔ ¯2-\(010 1‼⍠1 N M-1),M
</pre>

where <apll>‼⍠1</apll> uses the Variant operator <apll>⍠</apll> to evaluate <apll>‼</apll> in origin <apll>1</apll>.