View source for CombinatorialCase000/100

<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 <apll>⎕IO</apll>-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"
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{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">●</span>
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">●</span>
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">●</span>
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span>
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|&nbsp;&nbsp;&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|<span style="font-size: 2em;">&nbsp;</span>
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|&nbsp;&nbsp;&nbsp;
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as well as

{| border="0" cellpadding="5" cellspacing="0"
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{| border="1" cellpadding="5" cellspacing="0"
|❶
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|❷
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|❸
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|&nbsp;&nbsp;&nbsp;
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|&nbsp;
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{| border="1" cellpadding="5" cellspacing="0"
|&nbsp;&nbsp;&nbsp;
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|&nbsp;
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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>