46.7 Arrangements

Arrangements( mset )
Arrangements( mset, k )

NrArrangements( mset )
NrArrangements( mset, k )

In the first form Arrangements returns the set of arrangements of the multiset mset. In the second form Arrangements returns the set of all arrangements with k elements of the multiset mset.

In the first form NrArrangements returns the number of arrangements of the multiset mset. In the second form NrArrangements returns the number of arrangements with k elements of the multiset mset.

An arrangement of mset is an ordered selection without repetitions and is represented by a list that contains only elements from mset, but maybe in a different order. If mset is a proper set there are |mset|! / (|mset|-k)! (see Factorial) arrangements with k elements.

As an example of arrangements of a multiset, think of the game Scrabble. Suppose you have the six characters of the word settle and you have to make a four letter word. Then the possibilities are given by

    gap> Arrangements( ["s","e","t","t","l","e"], 4 );
    [ [ "e", "e", "l", "s" ], [ "e", "e", "l", "t" ],
      [ "e", "e", "s", "l" ], [ "e", "e", "s", "t" ],
      # 96 more possibilities
      [ "t", "t", "s", "e" ], [ "t", "t", "s", "l" ] ] 

Can you find the five proper English words, where lets does not count? Note that the fact that the list returned by Arrangements is a proper set means in this example that the possibilities are listed in the same order as they appear in the dictionary.

    gap> NrArrangements( ["s","e","t","t","l","e"] );
    523 

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

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