This case produces Partitions of the set {⍳M} into N ordered parts. Essentially, this case is the same as 102, except that the order of the elements is important so that there are more results by a factor of !N. For example, the 3-subset result of 1 2|3|4 for 102 is expanded to !4 (↔ 24) 3-subsets by permuting the values 1 2 3 4 in 24 ways.
- M labeled balls (1), N labeled boxes (1), at least one ball per box (2)
- Sensitive to ⎕IO
- Counted result is an integer scalar
- Generated result is a nested vector of nested integer vectors.
The count for this function is (!N)×M SN2 N where M SN2 N calculates the Stirling numbers of the 2nd kind..
For example:
If we have 4 labeled balls (❶❷❸❹) and 2 labeled boxes (12) with at least one ball per box, there are 14 (↔ (!2)×4 SN2 2 ↔ 2×7) ways to meet these criteria:
The diagram above corresponds to the nested array
⍪112 1‼4 2
1 2 3 4
4 1 2 3
1 2 4 3
3 1 2 4
1 2 3 4
3 4 1 2
1 3 4 2
2 1 3 4
1 3 2 4
2 4 1 3
1 4 2 3
2 3 1 4
1 2 3 4
2 3 4 1
⍝ Partitions of the set {⍳M} into
⍝ N ordered parts
⍝ Labeled balls & boxes, any # Balls per Box
⍪112 1‼3 3
1 2 3
2 1 3
2 3 1
1 3 2
3 1 2
3 2 1
⍪112 1‼3 2
1 2 3
3 1 2
1 3 2
2 1 3
1 2 3
2 3 1
⍪112 1‼3 1
1 2 3
In general, this case is equivalent to calculating the unlabeled boxes (102) and then permuting the items from that result as in
a←⊃102 1‼M N
b← 110 1‼N N
112 1‼M N ↔ ,⊂[⎕IO+2] a[;b]
or vice-versa
102 1‼M N ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼N N
