CombinatorialCase000/100

M Pigeons into N holes

Cases 000 and 100 are both trivial.

• M unlabeled (0) or labeled (1) balls into N 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 M≤N.

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

For example:

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

●

●

●

as well as

❶

❷

❸

The diagram above corresponds to

      000 1‼3 5
1 1 1 0 0

      ⍝ M pigeons into N holes
      ⍝ Unlabeled balls & boxes, ≤1 # Balls per Box
      000 1‼4 5
1 1 1 1 0