CombinatorialCase102 - Revision history

WikiSysop at 22:17, 14 May 2017

2017-05-14T22:17:34Z

← Older revision		Revision as of 18:17, 14 May 2017
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	This case produces '''Partitions of the set <apll>{⍳L}</apll> into exactly <apll>R</apll> parts'''. As such, it produces a subset of [[CombinatorialCase101\|<apll>101</apll>]], limiting the result to just those rows with <apll>L</apll> subsets.		This case produces '''Partitions of the set <apll>{⍳M}</apll> into exactly <apll>N</apll> parts'''. As such, it produces a subset of [[CombinatorialCase101\|<apll>101</apll>]], limiting the result to just those rows with <apll>M</apll> subsets.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)		* <apll>M</apll> labeled balls (1), <apll>N</apll> unlabeled boxes (0), at least one ball per box (2)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is a nested vector of nested integer vectors.		* Generated result is a nested vector of nested integer vectors.

	The count for this function is <apll>L SN2 R</apll> where <apll>L SN2 R</apll> calculates the [https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling numbers of the 2<sup>nd</sup> kind].		The count for this function is <apll>M SN2 N</apll> where <apll>M SN2 N</apll> calculates the [https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling numbers of the 2<sup>nd</sup> kind].

	For example:		For example:
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	1 2 3 4		1 2 3 4

	⍝ Partitions of {⍳L} into R parts		⍝ Partitions of {⍳M} into N parts
	⍝ Labeled balls, unlabeled boxes, ≥1 # Balls per Box		⍝ Labeled balls, unlabeled boxes, ≥1 # Balls per Box
	⍝ The number to the right in parens		⍝ The number to the right in parens
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	<pre>		<pre>
	101 ~~1‼L R~~ ↔ ⊃,/102 ~~1‼¨L~~,¨0..R		101 1‼M N ↔ ⊃,/102 1‼¨M,¨0..N
	102 ~~1‼L R~~ ↔ R {(⍺=≢¨⍵)/⍵} 101 ~~1‼L R~~		102 1‼M N ↔ R {(⍺=≢¨⍵)/⍵} 101 1‼M N
	</pre>		</pre>

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	<pre>		<pre>
	102 ~~1‼L R~~ ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 ~~1‼L R~~		102 1‼M N ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼M N
	a←⊃102 ~~1‼L R~~		a←⊃102 1‼M N
	b← 110 ~~1‼R R~~		b← 110 1‼M N
	112 ~~1‼L R~~ ↔ ,⊂[⎕IO+2] a[;b]		112 1‼M N ↔ ,⊂[⎕IO+2] a[;b]
	</pre>		</pre>

WikiSysop at 01:42, 30 April 2017

2017-04-30T01:42:54Z

← Older revision		Revision as of 21:42, 29 April 2017
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	This case produces '''Partitions of the set~~'''~~ <apll>{⍳L}</apll> into exactly <apll>R</apll> parts. As such, it produces a subset of [[CombinatorialCase101\|<apll>101</apll>]], limiting the result to just those rows with <apll>L</apll> subsets.		This case produces '''Partitions of the set <apll>{⍳L}</apll> into exactly <apll>R</apll> parts'''. As such, it produces a subset of [[CombinatorialCase101\|<apll>101</apll>]], limiting the result to just those rows with <apll>L</apll> subsets.

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

WikiSysop at 01:18, 30 April 2017

2017-04-30T01:18:09Z

← Older revision		Revision as of 21:18, 29 April 2017
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	⍝ Partitions of {⍳L} into R parts		⍝ Partitions of {⍳L} into R parts
	⍝ Labeled balls, unlabeled boxes, ≥1 #Balls per Box		⍝ Labeled balls, unlabeled boxes, ≥1 # Balls per Box
	⍝ The number to the right in parens		⍝ The number to the right in parens
	⍝ represent the corresponding row from		⍝ represent the corresponding row from

WikiSysop: Created page with "This case produces '''Partitions of the set''' {⍳L} into exactly R parts. As such, it produces a subset of CombinatorialCase101|101..."

2017-04-30T01:14:26Z

Created page with "This case produces '''Partitions of the set''' <apll>{⍳L}</apll> into exactly <apll>R</apll> parts. As such, it produces a subset of CombinatorialCase101|<apll>101</apll>..."

New page

This case produces '''Partitions of the set''' <apll>{⍳L}</apll> into exactly <apll>R</apll> parts. As such, it produces a subset of [[CombinatorialCase101|<apll>101</apll>]], limiting the result to just those rows with <apll>L</apll> subsets.

* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)
* Sensitive to <apll>⎕IO</apll>
* Counted result is an integer scalar
* Generated result is a nested vector of nested integer vectors.

The count for this function is <apll>L SN2 R</apll> where <apll>L SN2 R</apll> calculates the [https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling numbers of the 2nd kind].

For example:

If we have 4 labeled balls (❶❷❸❹) and 2 unlabeled boxes with at least one ball per box, there are <apll>7</apll> (<apll>↔ 4 SN2 2</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❷ ❸
|valign="bottom"|❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❷ ❹
|valign="bottom"|❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❶ ❷
|valign="bottom"|❸ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❸ ❹
|valign="bottom"|❷
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❶ ❸
|valign="bottom"|❷ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❶ ❹
|valign="bottom"|❷ ❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❶
|valign="bottom"|❷ ❸ ❹
|-
|   
|   
|}
|}

The diagram above corresponds to the nested array

<pre>
⍪102 1‼4 2
1 2 3 4
1 2 4 3
1 2 3 4
1 3 4 2
1 3 2 4
1 4 2 3
1 2 3 4

⍝ Partitions of {⍳L} into R parts
⍝ Labeled balls, unlabeled boxes, ≥1 #Balls per Box
⍝ The number to the right in parens
⍝ represent the corresponding row from
⍝ the table in case 101.

⍪102 1‼4 4
1 2 3 4 (15)
⍪102 1‼4 3
1 2 3 4 (5)
1 3 2 4 (8)
1 2 3 4 (11)
1 4 2 3 (12)
1 2 4 3 (13)
1 2 3 4 (14)
⍪102 1‼4 2
1 2 3 4 (2)
1 2 4 3 (3)
1 2 3 4 (4)
1 3 4 2 (6)
1 3 2 4 (7)
1 4 2 3 (9)
1 2 3 4 (10)
⍪102 1‼4 1
1 2 3 4 (1)
⍪102 1‼4 0
</pre>

In general, this case is related to [[CombinatorialCase101|<apll>101</apll>]] through the following identities (after sorting the items):

<pre>
101 1‼L R ↔ ⊃,/102 1‼¨L,¨0..R
102 1‼L R ↔ R {(⍺=≢¨⍵)/⍵} 101 1‼L R
</pre>

and is related to [[CombinatorialCase112|<apll>112</apll>]] through the following identities:

<pre>
102 1‼L R ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼L R
a←⊃102 1‼L R
b← 110 1‼R R
112 1‼L R ↔ ,⊂[⎕IO+2] a[;b]
</pre>