View source for CombinatorialCase101

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>M</apll> labeled balls (1), <apll>N</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>+/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:

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<br />❷<br />❸<br />❹
|&nbsp;&nbsp;&nbsp;
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br />❶<br />❷<br />❸
|valign="bottom"|❹
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br />❶<br />❷<br />❹
|valign="bottom"|❸
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br /><br />❶<br />❷
|valign="bottom"|❸<br />❹
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br />❶<br />❸<br />❹
|valign="bottom"|❷<br />
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br /><br />❶<br />❸
|valign="bottom"|❷<br />❹
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br /><br />❶<br />❹
|valign="bottom"|❷<br />❸
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|valign="bottom"|<br /><br /><br />❶
|valign="bottom"|❷<br />❸<br />❹
|-
|&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;
|}
|}

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 {⍳M} into at most N 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&nbsp;  4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 3&#124;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&nbsp;  3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2 4&#124;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&nbsp;  3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2&#124;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&nbsp; 3&nbsp; 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"| 1 2&#124;3&#124;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&nbsp; 2
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3 4&#124;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&nbsp; 2 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3&#124;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&nbsp; 2&nbsp; 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 3&#124;2&#124;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&nbsp; 2 3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 4&#124;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&nbsp; 2  3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1&#124;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&nbsp; 2 3&nbsp; 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1&#124;2 3&#124;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&nbsp; 2&nbsp;  3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1 4&#124;2&#124;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&nbsp; 2 4&nbsp; 3
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1&#124;2 4&#124;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&nbsp; 2&nbsp; 3 4
| style="border-top:0px; border-bottom:0px; padding-left: 2em; padding-right: 2em;"|1&#124;2&#124;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&nbsp; 2&nbsp; 3&nbsp; 4
| style="border-top:0px; border-bottom:inherit; padding-left: 2em; padding-right: 2em;"|1&#124;2&#124;3&#124;4
|}
 
See case [[CombinatorialCase102|<apll>102</apll>]] for identities that relate this case and [[CombinatorialCase102|<apll>102</apll>]].