CombinatorialCase012 - Revision history

WikiSysop at 21:50, 21 October 2017

2017-10-21T21:50:37Z

← Older revision		Revision as of 17:50, 21 October 2017
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	2 2 1		2 2 1
	3 1 1		3 1 1
	12 3‼5 3 ⍝ Gray Code order not implemented as yet		12 3‼5 3 ⍝ Gray Code order for Compositions not implemented as yet
	NONCE ERROR		NONCE ERROR
	12 3‼5 3		12 3‼5 3

WikiSysop at 21:07, 21 October 2017

2017-10-21T21:07:08Z

← Older revision		Revision as of 17:07, 21 October 2017
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	* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at least one ball per box (2)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at least one ball per box (2)
	* Not <apll>⎕IO</apll>-sensitive		* Not <apll>⎕IO</apll>-sensitive
			* Allows Lexicographic order
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is an integer matrix.		* Generated result is an integer matrix.
Line 78:		Line 79:

	<pre>		<pre>
	~~012~~ 1‼5 3		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 1 3
	1 2 2		1 2 2
Line 85:		Line 93:
	2 2 1		2 2 1
	3 1 1		3 1 1
			12 3‼5 3 ⍝ Gray Code order not implemented as yet
			NONCE ERROR
			12 3‼5 3
			∧

	⍝ Compositions of M into N parts		⍝ Compositions of M into N parts
	⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box		⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box
	~~012 1‼5~~ 5		12 2‼5 5
	1 1 1 1 1		1 1 1 1 1
	~~012 1‼5~~ 4		12 2‼5 4
	1 1 1 2		1 1 1 2
	1 1 2 1		1 1 2 1
	1 2 1 1		1 2 1 1
	2 1 1 1		2 1 1 1
	~~012 1‼5~~ 3		12 2‼5 3
	1 1 3		1 1 3
	1 2 2		1 2 2
Line 101:		Line 114:
	2 2 1		2 2 1
	3 1 1		3 1 1
	~~012 1‼5~~ 2		12 2‼5 2
	1 4		1 4
	2 3		2 3
	3 2		3 2
	4 1		4 1
	~~012 1‼5~~ 1		12 2‼5 1
	5		5
	</pre>		</pre>

WikiSysop at 23:51, 19 June 2017

2017-06-19T23:51:32Z

← Older revision		Revision as of 19:51, 19 June 2017
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	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 ~~partition~~ of <apll>M</apll> into <apll>N</apll> ~~ordered parts~~.		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)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), at least one ball per box (2)

WikiSysop at 22:10, 14 May 2017

2017-05-14T22:10:17Z

← Older revision		Revision as of 18:10, 14 May 2017
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	This case produces '''Compositions of the number <apll>L</apll> into <apll>R</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>L</apll> as the sum of <apll>R</apll> positive integers. It can also be thought of as a partition of <apll>L</apll> into <apll>R</apll> ordered parts.		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 partition of <apll>M</apll> into <apll>N</apll> ordered parts.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at least one ball per box (2)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> labeled boxes (1), 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-R)!L-1</apll>.		The count for this function is <apll>(M-N)!M-1</apll>.

	For example:		For example:
Line 85:		Line 85:
	2 2 1		2 2 1
	3 1 1		3 1 1
	⍝ Compositions of L into R parts		⍝ Compositions of M into N parts
	⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box		⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box
	012 1‼5 5		012 1‼5 5
Line 113:		Line 113:

	<pre>		<pre>
	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 01:45, 30 April 2017

2017-04-30T01:45:42Z

← Older revision		Revision as of 21:45, 29 April 2017
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	This case produces '''Compositions~~'''~~ of the number <apll>L</apll> into <apll>R</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>L</apll> as the sum of <apll>R</apll> positive integers. It can also be thought of as a partition of <apll>L</apll> into <apll>R</apll> ordered parts.		This case produces '''Compositions of the number <apll>L</apll> into <apll>R</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>L</apll> as the sum of <apll>R</apll> positive integers. It can also be thought of as a partition of <apll>L</apll> into <apll>R</apll> ordered parts.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at least one ball per box (2)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> labeled boxes (1), at least one ball per box (2)

WikiSysop at 01:17, 30 April 2017

2017-04-30T01:17:38Z

← Older revision		Revision as of 21:17, 29 April 2017
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	3 1 1		3 1 1
	⍝ Compositions of L into R parts		⍝ Compositions of L into R parts
	⍝ Unlabeled balls, labeled boxes, ≥1 #~~bpb~~		⍝ Unlabeled balls, labeled boxes, ≥1 # Balls per Box
	012 1‼5 5		012 1‼5 5
	1 1 1 1 1		1 1 1 1 1

WikiSysop: Created page with "This case produces '''Compositions''' of the number L into R parts. A composition is a way of representing a number as the sum of all positive integ..."

2017-04-29T22:11:22Z

Created page with "This case produces '''Compositions''' of the number <apll>L</apll> into <apll>R</apll> parts. A composition is a way of representing a number as the sum of all positive integ..."

New page

This case produces '''Compositions''' of the number <apll>L</apll> into <apll>R</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>L</apll> as the sum of <apll>R</apll> positive integers. It can also be thought of as a partition of <apll>L</apll> into <apll>R</apll> ordered parts.

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

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

For example:

If we have 5 unlabeled balls (●●●●●) 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"
|    ●
|    ●
|● ● ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|    ●
|  ● ●
|  ● ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|    ●
|● ● ●
|    ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|  ● ●
|    ●
|  ● ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|  ● ●
|  ● ●
|    ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|● ● ●
|    ●
|    ●
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|}

The diagram above corresponds to

<pre>
012 1‼5 3
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
⍝ Compositions of L into R parts
⍝ Unlabeled balls, labeled boxes, ≥1 #bpb
012 1‼5 5
1 1 1 1 1
012 1‼5 4
1 1 1 2
1 1 2 1
1 2 1 1
2 1 1 1
012 1‼5 3
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
012 1‼5 2
1 4
2 3
3 2
4 1
012 1‼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‼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>.