Lambda recursions on the example of generating permutations

[Robin Haberkorn]

Hello everybody,

I stumbled across an interesting little problem. I would like to generate all 
possible combinations of taking ⍺ things from a total of ⍵. The number of 
possible combinations is consequently ⍺!⍵.

Here's a straight-forward "brute force" solution:

Perm ← {(X/⍨⍺=+/¨X←(⊂2⍴⍨⍵)⊤¨⍳2*⍵) /¨ ⊂⍳⍵; X}

      2 Perm 4
 3 4  2 4  2 3  1 4  1 3  1 2
      3 Perm 4
 2 3 4  1 3 4  1 2 4  1 2 3

Unfortunately the algorithm has O(2^⍵) time and space complexity. Which is okay 
for what I am using it, but I was still curious of how this could be solved 
recursively in a lambda. I feel that simple things like this should have elegant 
solutions in GNU APL. Unfortunately this is the best I could come up with:

Perm ← {⍎∊('↑,/X,¨¨X+(⍺-1) Perm¨⍵-X' 'X')[1+⍺≤1] ⊣ X←⍳1+⍵-⍺; X}

At this point I was really feeling the limitations of lambdas in GNU APL. There 
apparently aren't enough means of control flow. At least I couldn't find a way 
to utilize one of APL's operators elegantly for this purpose, so I fell back to 
a computed ⍎, which feels really clumsy and is probably quite slow as well.

Or you write a traditional ∇ function with →... I mean you could also write your 
own ⍶IfElse⍹ operator, but due to the lack of lexical scoping this would be 
pretty useless for recursion, unless you pass in your context as ⍺ or ⍵ which is 
again very clumsy.

I would therefore claim that we are either lacking lexical scoping and/or 
built-in structured If-Else constructs. Or lambdas should simply be allowed to 
contain ⋄, new lines and →. Or I am missing something obvious. I am not sure. 
But IMHO something fundamental is missing in the language.

I am interested in hearing your thoughts.

Yours sincerely,
Robin

PS: I am not saying "I want it like in Dyalog!!!11". GNU APL is great for what 
it is and I am not ready to switch over to Dyalog yet. I particularly like two 
things about GNU APL: 1) It's FREE software. 2.) It's more conservatively 
designed than Dyalog.