WikiSysop at 22:28, 14 May 2017

2017-05-14T22:28:10Z

← Older revision		Revision as of 18:28, 14 May 2017
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	This case produces '''Partitions of the set <apll>{⍳L}</apll> into <apll>R</apll> ordered parts'''. Essentially, this case is the same as [[CombinatorialCase102\|<apll>102</apll>]], except that the order of the elements is important so that there are more results by a factor of <apll>!R</apll>. For example, the 3-subset result of <apll>1 2\|3\|4</apll> for [[CombinatorialCase102\|<apll>102</apll>]] is expanded to <apll>!4</apll> (<apll>↔ 24</apll>) 3-subsets by permuting the values <apll>1 2 3 4</apll> in <apll>24</apll> ways.		This case produces '''Partitions of the set <apll>{⍳M}</apll> into <apll>N</apll> ordered parts'''. Essentially, this case is the same as [[CombinatorialCase102\|<apll>102</apll>]], except that the order of the elements is important so that there are more results by a factor of <apll>!N</apll>. For example, the 3-subset result of <apll>1 2\|3\|4</apll> for [[CombinatorialCase102\|<apll>102</apll>]] is expanded to <apll>!4</apll> (<apll>↔ 24</apll>) 3-subsets by permuting the values <apll>1 2 3 4</apll> in <apll>24</apll> ways.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at least one ball per box (2)		* <apll>M</apll> labeled balls (1), <apll>N</apll> labeled boxes (1), 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>(!R)×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>(!N)×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
	2 3 4 1		2 3 4 1
	⍝ Partitions of the set {⍳L} into		⍝ Partitions of the set {⍳M} into
	⍝ R ordered parts		⍝ N ordered parts
	⍝ Labeled balls & boxes, any # Balls per Box		⍝ Labeled balls & boxes, any # Balls per Box
	⍪112 1‼3 3		⍪112 1‼3 3
Line 171:		Line 171:

	<pre>		<pre>
	a←⊃102 ~~1‼L R~~		a←⊃102 1‼M N
	b← 110 ~~1‼R R~~		b← 110 1‼N N
	112 ~~1‼L R~~ ↔ ,⊂[⎕IO+2] a[;b]		112 1‼M N ↔ ,⊂[⎕IO+2] a[;b]
	</pre>		</pre>

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	<pre>		<pre>
	102 ~~1‼L R~~ ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 ~~1‼R R~~		102 1‼M N ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼N N
	</pre>		</pre>

WikiSysop: Created page with "This case produces '''Partitions of the set {⍳L} into R ordered parts'''. Essentially, this case is the same as CombinatorialCase102|102
2017-04-30T01:42:14Z

Created page with "This case produces '''Partitions of the set <apll>{⍳L}</apll> into <apll>R</apll> ordered parts'''. Essentially, this case is the same as CombinatorialCase102|<apll>102</..."

New page
This case produces '''Partitions of the set <apll>{⍳L}</apll> into <apll>R</apll> ordered parts'''. Essentially, this case is the same as [[CombinatorialCase102|<apll>102</apll>]], except that the order of the elements is important so that there are more results by a factor of <apll>!R</apll>. For example, the 3-subset result of <apll>1 2|3|4</apll> for [[CombinatorialCase102|<apll>102</apll>]] is expanded to <apll>!4</apll> (<apll>↔ 24</apll>) 3-subsets by permuting the values <apll>1 2 3 4</apll> in <apll>24</apll> ways.

* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), 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>(!R)×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 labeled boxes (<apll>12</apll>) with at least one ball per box, there are <apll>14</apll> (<apll>↔ (!2)×4 SN2 2 ↔ 2×7</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"|❹
|valign="bottom"|❶ ❷ ❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❷ ❹
|valign="bottom"|❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❸
|valign="bottom"|❶ ❷ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❶ ❷
|valign="bottom"|❸ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❸ ❹
|valign="bottom"|❶ ❷
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❸ ❹
|valign="bottom"|❷
|-
|   
|   
|}
|}
{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❷
|valign="bottom"|❶ ❸ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❸
|valign="bottom"| ❷ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❷ ❹
|valign="bottom"|❶ ❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶ ❹
|valign="bottom"| ❷ ❸
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"| ❷ ❸
|valign="bottom"|❶ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❶
|valign="bottom"|❷ ❸ ❹
|-
|   
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|❷ ❸ ❹
|valign="bottom"|❶
|-
|   
|   
|}
|}

The diagram above corresponds to the nested array

<pre>
⍪112 1‼4 2
1 2 3 4
4 1 2 3
1 2 4 3
3 1 2 4
1 2 3 4
3 4 1 2
1 3 4 2
2 1 3 4
1 3 2 4
2 4 1 3
1 4 2 3
2 3 1 4
1 2 3 4
2 3 4 1
⍝ Partitions of the set {⍳L} into
⍝ R ordered parts
⍝ Labeled balls & boxes, any # Balls per Box
⍪112 1‼3 3
1 2 3
2 1 3
2 3 1
1 3 2
3 1 2
3 2 1
⍪112 1‼3 2
1 2 3
3 1 2
1 3 2
2 1 3
1 2 3
2 3 1
⍪112 1‼3 1
1 2 3
</pre>

In general, this case is equivalent to calculating the unlabeled boxes ([[CombinatorialCase102|<apll>102</apll>]]) and then permuting the items from that result as in

<pre>
a←⊃102 1‼L R
b← 110 1‼R R
112 1‼L R ↔ ,⊂[⎕IO+2] a[;b]
</pre>

or vice-versa

<pre>
102 1‼L R ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼R R
</pre>

CombinatorialCase112 - Revision history

WikiSysop at 22:28, 14 May 2017