How to generate random combinations from large set in GAP

Question

I need to generate random combinations (unordered tuples without duplication). This code works fine for relatively small $n$ but I need much larger n $n>200$ for example

n:=20; # fails for n=200
enum:=EnumeratorOfCombinations([1..n]);
for k in [1..10] do
 b:=enum[Random([1..2^n])];
 Print(b,"\n");
od;

# result :
[ 1, 2, 3, 5, 9, 11, 14, 15, 17, 18, 20 ]
[ 1, 5, 6, 7, 9, 11, 12, 14, 17, 18 ]
[ 2, 4, 5, 6, 10, 13, 14, 18, 19 ]
[ 2, 7, 9, 10, 11, 12, 14, 17, 18 ]
[ 3, 9, 10, 11, 13, 14, 15, 17, 18 ]
[ 3, 5, 7, 8, 9, 11, 14, 17, 18, 19, 20 ]
[ 1, 2, 3, 6, 7, 10, 15, 16, 19 ]
[ 1, 2, 4, 7, 8, 10, 12, 13, 17, 19, 20 ]
[ 1, 2, 3, 7, 8, 13, 14, 17 ]
[ 2, 3, 4, 6, 8, 16, 17, 18, 19 ]

Are there other options for doing this in GAP?

Answer 1

Do you really want to produce random combinations of any size (instead of some fixed size)? So, if for example $n=3$, you want to pick uniformly at random from the 8 combinations [], [1], [2], [3], [1,2], [1,3], [2,3], [1,2,3]? That's an incredibly convoluted way of saying you want to pick a random subset of the $n$ elements, with all $2^n$ subsets equally probable. I'm sure you can find easier ways of doing that. (For each of the $n$ elements, you just want to include it in the subset with probability $0.5$.)

That having been said, your interesting solution does almost work. For large $n$ it fails because Random([1..2^n]) tries to build a Range with a huge upper limit:

gap> n:=200;; [1..2^n];
Error, Range: <last> must be a small integer (not a large positive integer)

You can instead call a version of Random that does not require a range. Just give it the lower and upper limits as integers.

gap> enum:=EnumeratorOfCombinations([1..200]);;
gap> n := 200;; b := Random(1, 2^n); enum[b];
435499391128194516091289418442286066942494750160332640859166
[ 1, 3, 4, 5, 11, 13, 14, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 30, 31, 35, 39, 42, 43, 44, 45, 46, 48, 49, 52, 54,
  55, 57, 61, 62, 64, 65, 70, 74, 75, 76, 77, 78, 79, 80, 81, 87, 88, 89, 90, 93, 94, 96, 97, 98, 99, 103, 104, 107,
  108, 110, 111, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 129, 130, 131, 133, 134, 135, 136, 137,
  140, 141, 143, 144, 146, 148, 149, 150, 151, 153, 154, 159, 160, 161, 162, 163, 164, 165, 167, 168, 169, 170, 172,
  175, 176, 177, 180, 185, 190, 191, 193, 195, 199 ]