CombinatorialCase000/100 - Revision history

WikiSysop at 16:09, 14 May 2017

2017-05-14T16:09:40Z

← Older revision		Revision as of 12:09, 14 May 2017
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	<apll>L</apll> Pigeons into <apll>R</apll> holes		<apll>M</apll> Pigeons into <apll>N</apll> holes

	Cases '''000''' and '''100''' are both trivial.		Cases '''000''' and '''100''' are both trivial.

	* <apll>L</apll> unlabeled (0) or labeled (1) balls into <apll>R</apll> unlabeled boxes (0), at most one per box (0)		* <apll>M</apll> unlabeled (0) or labeled (1) balls into <apll>N</apll> unlabeled boxes (0), at most one per box (0)
	* Not <apll>⎕IO</apll>-sensitive		* Not <apll>⎕IO</apll>-sensitive
	* Counted result is a Boolean scalar		* Counted result is a Boolean scalar
	* Generated result is a Boolean matrix.		* Generated result is a Boolean matrix.

	The count for this function is <apll>~~L≤R~~</apll>.		The count for this function is <apll>M≤N</apll>.

	If <apll>~~L>R~~</apll>, then there is no answer, or more accurately, the result is an empty matrix of shape <apll>0 R</apll>. If <apll>~~L≤R~~</apll>, then the result is a one-row matrix with <apll>L</apll> leading <apll>1</apll>s and the rest <apll>0</apll>s. Combining these two cases yields a result of <apll>((~~L≤R~~) R)~~⍴R↑L⍴1~~</apll>.		If <apll>M>N</apll>, then there is no answer, or more accurately, the result is an empty matrix of shape <apll>0 N</apll>. If <apll>M≤N</apll>, then the result is a one-row matrix with <apll>M</apll> leading <apll>1</apll>s and the rest <apll>0</apll>s. Combining these two cases yields a result of <apll>((M≤N) N)⍴N↑M⍴1</apll>.

	For example:		For example:
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	1 1 1 0 0		1 1 1 0 0

	⍝ L pigeons into R holes		⍝ M pigeons into N holes
	⍝ Unlabeled balls & boxes, ≤1 # Balls per Box		⍝ Unlabeled balls & boxes, ≤1 # Balls per Box
	000 1‼4 5		000 1‼4 5
	1 1 1 1 0</pre></apll>		1 1 1 1 0</pre></apll>

WikiSysop at 01:58, 30 April 2017

2017-04-30T01:58:33Z

← Older revision		Revision as of 21:58, 29 April 2017
Line 91:		Line 91:

	⍝ L pigeons into R holes		⍝ L pigeons into R holes
	⍝ Unlabeled balls & boxes, ≤1 #~~bpb~~		⍝ Unlabeled balls & boxes, ≤1 # Balls per Box
	000 1‼4 5		000 1‼4 5
	1 1 1 1 0</pre></apll>		1 1 1 1 0</pre></apll>

WikiSysop at 18:21, 29 April 2017

2017-04-29T18:21:37Z

← Older revision		Revision as of 14:21, 29 April 2017
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	* <apll>L</apll> unlabeled (0) or labeled (1) balls into <apll>R</apll> unlabeled boxes (0), at most one per box (0)		* <apll>L</apll> unlabeled (0) or labeled (1) balls into <apll>R</apll> unlabeled boxes (0), at most one per box (0)
	* Not ⎕IO-sensitive		* Not <apll>⎕IO</apll>-sensitive
	* Counted result is a Boolean scalar		* Counted result is a Boolean scalar
	* Generated result is a Boolean matrix.		* Generated result is a Boolean matrix.

WikiSysop: Created page with "L Pigeons into R holes Cases '''000''' and '''100''' are both trivial. * L unlabeled (0) or labeled (1) balls into R unla..."

2017-04-29T15:31:54Z

Created page with "<apll>L</apll> Pigeons into <apll>R</apll> holes Cases '''000''' and '''100''' are both trivial. * <apll>L</apll> unlabeled (0) or labeled (1) balls into <apll>R</apll> unla..."

New page

<apll>L</apll> Pigeons into <apll>R</apll> holes

Cases '''000''' and '''100''' are both trivial.

* <apll>L</apll> unlabeled (0) or labeled (1) balls into <apll>R</apll> unlabeled boxes (0), at most one per box (0)
* Not ⎕IO-sensitive
* Counted result is a Boolean scalar
* Generated result is a Boolean matrix.

The count for this function is <apll>L≤R</apll>.

If <apll>L>R</apll>, then there is no answer, or more accurately, the result is an empty matrix of shape <apll>0 R</apll>. If <apll>L≤R</apll>, then the result is a one-row matrix with <apll>L</apll> leading <apll>1</apll>s and the rest <apll>0</apll>s. Combining these two cases yields a result of <apll>((L≤R) R)⍴R↑L⍴1</apll>.

For example:

If we have <apll>3</apll> unlabeled balls (●●●) and <apll>5</apll> unlabeled boxes with at most one ball per box, there is only <apll>1</apll> (<apll>↔ 3≤5</apll>) way to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|●
|-
| 
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|●
|-
| 
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|●
|-
| 
|}
|
{| border="1" cellpadding="5" cellspacing="0"
| 
|-
|   
|}
|
{| border="1" cellpadding="5" cellspacing="0"
| 
|-
|   
|}
|}

as well as

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

The diagram above corresponds to

<apll><pre>
000 1‼3 5
1 1 1 0 0

⍝ L pigeons into R holes
⍝ Unlabeled balls & boxes, ≤1 #bpb
000 1‼4 5
1 1 1 1 0</pre></apll>