view release on metacpan or  search on metacpan

Combinatorics.pm  view on Meta::CPAN

    }

The C<next()> method returns an arrayref to the next tuple, if any, or C<undef> if the
sequence is exhausted.

Memory usage is minimal, no recursion and no stacks are involved.

=item *

In list context subroutines slurp the entire set of tuples. This behaviour is offered
for convenience, but take into account that the resulting array may be really huge:

    my @all_combinations = combinations(\@data, $k);

=back


=head2 permutations(\@data)

The permutations of C<@data> are all its reorderings. For example, the permutations of C<@data = (1, 2, 3)> are:

    (1, 2, 3)
    (1, 3, 2)
    (2, 1, 3)
    (2, 3, 1)
    (3, 1, 2)
    (3, 2, 1)

The number of permutations of C<n> elements is:

    n! = 1,                  if n = 0
    n! = n*(n-1)*...*1,      if n > 0

See some values at L<http://www.research.att.com/~njas/sequences/A000142>.


=head2 circular_permutations(\@data)

The circular permutations of C<@data> are its arrangements around a circle, where only relative order of elements matter, rather than their actual position. Think possible arrangements of people around a circular table for dinner according to whom th...

For example the circular permutations of C<@data = (1, 2, 3, 4)> are:

    (1, 2, 3, 4)
    (1, 2, 4, 3)
    (1, 3, 2, 4)
    (1, 3, 4, 2)
    (1, 4, 2, 3)
    (1, 4, 3, 2)

The number of circular permutations of C<n> elements is:

        n! = 1,                      if 0 <= n <= 1
    (n-1)! = (n-1)*(n-2)*...*1,      if n > 1

See a few numbers in a comment of L<http://www.research.att.com/~njas/sequences/A000142>.


=head2 derangements(\@data)

The derangements of C<@data> are those reorderings that have no element
in its original place. In jargon those are the permutations of C<@data>
with no fixed points. For example, the derangements of C<@data = (1, 2,
3)> are:

    (2, 3, 1)
    (3, 1, 2)

The number of derangements of C<n> elements is:

    d(n) = 1,                       if n = 0
    d(n) = n*d(n-1) + (-1)**n,      if n > 0

See some values at L<http://www.research.att.com/~njas/sequences/A000166>.


=head2 complete_permutations(\@data)

This is an alias for C<derangements>, documented above.


=head2 variations(\@data, $k)

The variations of length C<$k> of C<@data> are all the tuples of length C<$k> consisting of elements of C<@data>. For example, for C<@data = (1, 2, 3)> and C<$k = 2>:

    (1, 2)
    (1, 3)
    (2, 1)
    (2, 3)
    (3, 1)
    (3, 2)

For this to make sense, C<$k> has to be less than or equal to the length of C<@data>.

Note that

    permutations(\@data);

is equivalent to

    variations(\@data, scalar @data);

The number of variations of C<n> elements taken in groups of C<k> is:

    v(n, k) = 1,                        if k = 0
    v(n, k) = n*(n-1)*...*(n-k+1),      if 0 < k <= n


=head2 variations_with_repetition(\@data, $k)

The variations with repetition of length C<$k> of C<@data> are all the tuples of length C<$k> consisting of elements of C<@data>, including repetitions. For example, for C<@data = (1, 2, 3)> and C<$k = 2>:

    (1, 1)
    (1, 2)
    (1, 3)
    (2, 1)
    (2, 2)
    (2, 3)
    (3, 1)
    (3, 2)
    (3, 3)