46.9 Tuples

Tuples( set, k )

NrTuples( set, k )

Tuples returns the set of all ordered tuples of length k of the set set.

NrTuples returns the number of all ordered tuples of length k of the set set.

An ordered tuple of length k of set is an ordered selection with repetition and is represented by a list of length k containing elements of set. There are |set|^k such ordered tuples.

Note that the fact that Tuples returns a set implies that the last index runs fastest. That means the first tuple contains the smallest element from set k times, the second tuple contains the smallest element of set at all positions except at the last positions, where it contains the second smallest element from set and so on.

    gap> Tuples( [1,2,3], 2 );
    [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], 
      [ 3, 1 ], [ 3, 2 ], [ 3, 3 ] ]
    gap> NrTuples( [1..10], 5 );
    100000 

Tuples(set,k) can also be viewed as the k-fold cartesian product of set (see Cartesian).

The function Combinations (see Combinations) computes unordered selections without repetitions, Arrangements (see Arrangements) computes ordered selections without repetitions, and finally the function UnorderedTuples (see UnorderedTuples) computes unordered selections with repetitions.

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