CombinatorialCase010 - Revision history

WikiSysop at 20:54, 21 October 2017

2017-10-21T20:54:31Z

@@ Line 3: / Line 3: @@
 * <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at most one ball per box (0)
 * Sensitive to <apll>⎕IO</apll>
 * Count result is an integer scalar
 * Generated result is an integer matrix.
@@ Line 92: / Line 93: @@
 <pre>
-1‼2 4
+1‼2 4 ⍝ Combinations in unspecified order
 2
 3
@@ Line 99: / Line 107: @@
 4
 4
        ⍝ M Combinations of N items
        ⍝ Unlabeled balls, labeled boxes, ≤1 # Balls per Box
@@ Line 109: / Line 124: @@
        ⍴010 1‼3 5
 3
-1‼3 5
+1‼3 5 ⍝ Combinations in unspecified order
 2 3
 2 4
@@ Line 120: / Line 146: @@
 4 5
 4 5
 </pre>

WikiSysop at 22:03, 14 May 2017

2017-05-14T22:03:08Z

← Older revision		Revision as of 18:03, 14 May 2017
Line 1:		Line 1:
	This case produces the '''<apll>L</apll> Combinations of <apll>R</apll> items'''.		This case produces the '''<apll>M</apll> Combinations of <apll>N</apll> items'''.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at most one ball per box (0)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
	* Count result is an integer scalar		* Count 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!R</apll>.		The count for this function is <apll>M!N</apll>.

	For example:		For example:
Line 87:		Line 87:
	\|}		\|}

	and, in general, it’s easy to see that this case solves the familiar problem of <apll>L</apll> combinations of <apll>R</apll> items.		and, in general, it’s easy to see that this case solves the familiar problem of <apll>M</apll> combinations of <apll>N</apll> items.

	The diagram above corresponds to		The diagram above corresponds to
Line 99:		Line 99:
	2 4		2 4
	3 4		3 4
	⍝ Combinations		⍝ M Combinations of N items
	⍝ Unlabeled balls, labeled boxes, ≤1 # Balls per Box		⍝ Unlabeled balls, labeled boxes, ≤1 # Balls per Box
	3!5		3!5
Line 125:		Line 125:

	<pre>		<pre>
	010 ~~1‼L R~~ ↔ (011 ~~1‼L~~,R-L-1)+[⎕IO+1] 0..L-1		010 1‼M N ↔ (011 1‼M,N-M-1)+[⎕IO+1] 0..M-1
	011 ~~1‼L R~~ ↔ (010 ~~1‼L~~,L+R-1)-[⎕IO+1] 0..L-1		011 1‼M N ↔ (010 1‼M,M+N-1)-[⎕IO+1] 0..M-1

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

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

WikiSysop at 01:44, 30 April 2017

2017-04-30T01:44:37Z

← Older revision		Revision as of 21:44, 29 April 2017
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	This case produces the <apll>L</apll> ~~'''~~Combinations~~'''~~ of <apll>R</apll> items.		This case produces the '''<apll>L</apll> Combinations of <apll>R</apll> items'''.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)

WikiSysop at 01:16, 30 April 2017

2017-04-30T01:16:39Z

← Older revision		Revision as of 21:16, 29 April 2017
Line 100:		Line 100:
	3 4		3 4
	⍝ Combinations		⍝ Combinations
	⍝ Unlabeled balls, labeled boxes, ≤1 #~~bpb~~		⍝ Unlabeled balls, labeled boxes, ≤1 # Balls per Box
	3!5		3!5
	10		10

WikiSysop at 22:38, 29 April 2017

2017-04-29T22:38:50Z

← Older revision		Revision as of 18:38, 29 April 2017
Line 1:		Line 1:
	This case produces the <apll>L</apll> ~~combinations~~ of <apll>R</apll> items.		This case produces the <apll>L</apll> '''Combinations''' of <apll>R</apll> items.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)

WikiSysop: Created page with "This case produces the L combinations of R items. * L unlabeled balls (0), R labeled boxes (1), at most one ball per box (..."

2017-04-29T21:17:50Z

Created page with "This case produces the <apll>L</apll> combinations of <apll>R</apll> items. * <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (..."

New page

This case produces the <apll>L</apll> combinations of <apll>R</apll> items.

* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at most one ball per box (0)
* Sensitive to <apll>⎕IO</apll>
* Count result is an integer scalar
* Generated result is an integer matrix.

The count for this function is <apll>L!R</apll>.

For example:

If we have 2 unlabeled balls (●●) and 4 labeled boxes (<apll>1234</apll>) with at most one ball per box, there are <apll>6</apll> (<apll>↔ 2!4</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
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{| border="1" cellpadding="5" cellspacing="0"
|●
|●
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|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
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{| border="1" cellpadding="5" cellspacing="0"
|●
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|●
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|-
|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
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{| border="1" cellpadding="5" cellspacing="0"
|●
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|●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
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{| border="1" cellpadding="5" cellspacing="0"
|   
|●
|●
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|-
|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
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{| border="1" cellpadding="5" cellspacing="0"
|   
|●
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|●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
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{| border="1" cellpadding="5" cellspacing="0"
|   
|   
|●
|●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|align="center"|4
|}
|}

and, in general, it’s easy to see that this case solves the familiar problem of <apll>L</apll> combinations of <apll>R</apll> items.

The diagram above corresponds to

<pre>
010 1‼2 4
1 2
1 3
1 4
2 3
2 4
3 4
⍝ Combinations
⍝ Unlabeled balls, labeled boxes, ≤1 #bpb
3!5
10
010‼3 5
10
010 0‼3 5
10
⍴010 1‼3 5
10 3
010 1‼3 5
1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5
</pre>

In general, this case is related to that of Multisets ([[CombinatorialCase011|<apll>011</apll>]]) and Compositions ([[CombinatorialCase012|<apll>012</apll>]]) via the following identities:

<pre>
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

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

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