WikiSysop at 22:13, 14 May 2017

2017-05-14T22:13:21Z

← Older revision		Revision as of 18:13, 14 May 2017
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	This case produces '''Partitions of the set <apll>{⍳L}</apll> into at most <apll>R</apll> parts'''. Partitioning a set is different from partitioning a number in that the former subdivides the set into non-empty disjoint subsets.		This case produces '''Partitions of the set <apll>{⍳M}</apll> into at most <apll>N</apll> parts'''. Partitioning a set is different from partitioning a number in that the former subdivides the set into non-empty disjoint subsets.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), any # balls per box (1)		* <apll>M</apll> labeled balls (1), <apll>N</apll> unlabeled boxes (0), any # balls per box (1)
	* 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¨0..R</apll> where <apll>L SN2 R</apll> returns 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¨0..N</apll> where <apll>M SN2 N</apll> returns 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:
Line 91:		Line 91:
	1 4 2 3		1 4 2 3
	1 2 3 4		1 2 3 4
	⍝ Partitions of {⍳L} into at most R parts		⍝ Partitions of {⍳M} into at most N parts
	⍝ Labeled balls, unlabeled boxes, any # Balls per Box		⍝ Labeled balls, unlabeled boxes, any # Balls per Box
	101 0‼4 4		101 0‼4 4

WikiSysop at 01:46, 30 April 2017

2017-04-30T01:46:19Z

← Older revision		Revision as of 21:46, 29 April 2017
Line 1:		Line 1:
	This case produces '''Partitions of the set~~'''~~ <apll>{⍳L}</apll> into at most <apll>R</apll> parts. Partitioning a set is different from partitioning a number in that the former subdivides the set into non-empty disjoint subsets.		This case produces '''Partitions of the set <apll>{⍳L}</apll> into at most <apll>R</apll> parts'''. Partitioning a set is different from partitioning a number in that the former subdivides the set into non-empty disjoint subsets.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), any # balls per box (1)		* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), any # balls per box (1)

WikiSysop at 01:02, 30 April 2017

2017-04-30T01:02:19Z

← Older revision		Revision as of 21:02, 29 April 2017
Line 6:		Line 6:
	* 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¨0..R</apll> where <apll>L SN2 R</apll> returns the Stirling numbers of the 2<sup>nd</sup> kind.		The count for this function is <apll>+/L SN2¨0..R</apll> where <apll>L SN2 R</apll> returns 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:

WikiSysop: Created page with "This case produces '''Partitions of the set''' {⍳L} into at most R parts. Partitioning a set is different from partitioning a number in that the f..."

2017-04-30T00:50:02Z

Created page with "This case produces '''Partitions of the set''' <apll>{⍳L}</apll> into at most <apll>R</apll> parts. Partitioning a set is different from partitioning a number in that the f..."

New page

This case produces '''Partitions of the set''' <apll>{⍳L}</apll> into at most <apll>R</apll> parts. Partitioning a set is different from partitioning a number in that the former subdivides the set into non-empty disjoint subsets.

* <apll>L</apll> labeled balls (1), <apll>R</apll> unlabeled boxes (0), any # balls per box (1)
* 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¨0..R</apll> where <apll>L SN2 R</apll> returns the Stirling numbers of the 2nd kind.

For example:

If we have 4 labeled balls (❶❷❸❹) and 2 unlabeled boxes with any # of balls per box, there are <apll>8</apll> (<apll>↔ +/4 SN2¨0..2 ↔ +/0 1 7</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|❶ ❷ ❸ ❹
|   
|-
|   
|   
|}
|
{| 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>
⍪101 1‼4 2
1 2 3 4
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 at most R parts
⍝ Labeled balls, unlabeled boxes, any # Balls per Box
101 0‼4 4
15
101 0‼4 3
14
101 0‼4 2
8
101 0‼4 1
1
101 0‼4 0
0
</pre>

The first column in the following table serves to number the rows and is referenced in [[CombinatorialCase102|<apll>102</apll>]]. The second column illustrates the result of <apll>101 1‼4 4</apll> as a nested array with <apll>15</apll> elements. The third column includes visual separators to make it clearer where one subset stops and another begins:

{| border="1" cellpadding="5" cellspacing="0" style='font-family: "SImPL medium", "SImPL", "APL385 Unicode", "APLX Upright", monospace;'
| style="border-top: inherit; border-bottom: 0px; text-align: right; padding-right: 1em;"|1
| style="border-top: inherit; border-bottom: 0px; padding-left: 2em; padding-right: 2em;"| 1 2 3 4
| style="border-top: inherit; border-bottom: 0px; padding-left: 2em; padding-right: 2em;"| 1 2 3 4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|2
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 3  4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 3|4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 4  3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 4|3
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2  3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2|3 4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|5
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2  3  4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2|3|4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|6
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 3 4  2
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3 4|2
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|7
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 3  2 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3|2 4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|8
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 3  2  4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3|2|4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|9
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 4  2 3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 4|2 3
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|10
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1  2 3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1|2 3 4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|11
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1  2 3  4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1|2 3|4
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|12
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 4  2  3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 4|2|3
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|13
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1  2 4  3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1|2 4|3
|-
| style="border-top:0px; border-bottom:0px; text-align: right; padding-right: 1em;"|14
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1  2  3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1|2|3 4
|-
| style="border-top:0px; border-bottom:inherit; text-align: right; padding-right: 1em;"|15
| style="border-top:0px; border-bottom:inherit; padding-left: 2em; padding-right: 2em;"| 1  2  3  4
| style="border-top:0px; border-bottom:inherit; padding-left: 2em; padding-right: 2em;"|1|2|3|4
|}

See case [[CombinatorialCase102|<apll>102</apll>]] for identities that relate this case and [[CombinatorialCase102|<apll>102</apll>]].

CombinatorialCase101 - Revision history

WikiSysop at 22:13, 14 May 2017

WikiSysop at 01:46, 30 April 2017

WikiSysop at 01:02, 30 April 2017

WikiSysop: Created page with "This case produces '''Partitions of the set''' {⍳L} into at most R parts. Partitioning a set is different from partitioning a number in that the f..."