# NAME

Algorithm::Combinatorics - Efficient generation of combinatorial sequences

# SYNOPSIS

`````` use Algorithm::Combinatorics qw(permutations);

my @data = qw(a b c);

# scalar context gives an iterator
my \$iter = permutations(\@data);
while (my \$p = \$iter->next) {
# ...
}

# list context slurps
my @all_permutations = permutations(\@data);``````

# VERSION

This documentation refers to Algorithm::Combinatorics version 0.26.

# DESCRIPTION

Algorithm::Combinatorics is an efficient generator of combinatorial sequences. Algorithms are selected from the literature (work in progress, see "REFERENCES"). Iterators do not use recursion, nor stacks, and are written in C.

Tuples are generated in lexicographic order, except in `subsets()`.

# SUBROUTINES

Algorithm::Combinatorics provides these subroutines:

``````    permutations(\@data)
circular_permutations(\@data)
derangements(\@data)
complete_permutations(\@data)
variations(\@data, \$k)
variations_with_repetition(\@data, \$k)
tuples(\@data, \$k)
tuples_with_repetition(\@data, \$k)
combinations(\@data, \$k)
combinations_with_repetition(\@data, \$k)
partitions(\@data[, \$k])
subsets(\@data[, \$k])``````

All of them are context-sensitive:

• In scalar context subroutines return an iterator that responds to the `next()` method. Using this object you can iterate over the sequence of tuples one by one this way:

``````    my \$iter = combinations(\@data, \$k);
while (my \$c = \$iter->next) {
# ...
}``````

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

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

• 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);``

## permutations(\@data)

The permutations of `@data` are all its reorderings. For example, the permutations of `@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 `n` elements is:

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

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

## circular_permutations(\@data)

The circular permutations of `@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 they have to their right and left, no matter the actual chair they sit on.

For example the circular permutations of `@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 `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 http://www.research.att.com/~njas/sequences/A000142.

## derangements(\@data)

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

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

The number of derangements of `n` elements is:

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

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

## complete_permutations(\@data)

This is an alias for `derangements`, documented above.

## variations(\@data, \$k)

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

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

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

Note that

``    permutations(\@data);``

is equivalent to

``    variations(\@data, scalar @data);``

The number of variations of `n` elements taken in groups of `k` is:

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

## variations_with_repetition(\@data, \$k)

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

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

Note that `\$k` can be greater than the length of `@data`. For example, for `@data = (1, 2)` and `\$k = 3`:

``````    (1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 2, 2)
(2, 1, 1)
(2, 1, 2)
(2, 2, 1)
(2, 2, 2)``````

The number of variations with repetition of `n` elements taken in groups of `k >= 0` is:

``    vr(n, k) = n**k``

## tuples(\@data, \$k)

This is an alias for `variations`, documented above.

## tuples_with_repetition(\@data, \$k)

This is an alias for `variations_with_repetition`, documented above.

## combinations(\@data, \$k)

The combinations of length `\$k` of `@data` are all the sets of size `\$k` consisting of elements of `@data`. For example, for `@data = (1, 2, 3, 4)` and `\$k = 3`:

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

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

The number of combinations of `n` elements taken in groups of `0 <= k <= n` is:

``    n choose k = n!/(k!*(n-k)!)``

## combinations_with_repetition(\@data, \$k);

The combinations of length `\$k` of an array `@data` are all the bags of size `\$k` consisting of elements of `@data`, with repetitions. For example, for `@data = (1, 2, 3)` and `\$k = 2`:

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

Note that `\$k` can be greater than the length of `@data`. For example, for `@data = (1, 2, 3)` and `\$k = 4`:

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

The number of combinations with repetition of `n` elements taken in groups of `k >= 0` is:

``    n+k-1 over k = (n+k-1)!/(k!*(n-1)!)``

## partitions(\@data[, \$k])

A partition of `@data` is a division of `@data` in separate pieces. Technically that's a set of subsets of `@data` which are non-empty, disjoint, and whose union is `@data`. For example, the partitions of `@data = (1, 2, 3)` are:

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

This subroutine returns in consequence tuples of tuples. The top-level tuple (an arrayref) represents the partition itself, whose elements are tuples (arrayrefs) in turn, each one representing a subset of `@data`.

The number of partitions of a set of `n` elements are known as Bell numbers, and satisfy the recursion:

``````    B(0) = 1
B(n+1) = (n over 0)B(0) + (n over 1)B(1) + ... + (n over n)B(n)``````

See some values at http://www.research.att.com/~njas/sequences/A000110.

If you pass the optional parameter `\$k`, the subroutine generates only partitions of size `\$k`. This uses an specific algorithm for partitions of known size, which is more efficient than generating all partitions and filtering them by size.

Note that in that case the subsets themselves may have several sizes, it is the number of elements of the partition which is `\$k`. For instance if `@data` has 5 elements there are partitions of size 2 that consist of a subset of size 2 and its complement of size 3; and partitions of size 2 that consist of a subset of size 1 and its complement of size 4. In both cases the partitions have the same size, they have two elements.

The number of partitions of size `k` of a set of `n` elements are known as Stirling numbers of the second kind, and satisfy the recursion:

``````    S(0, 0) = 1
S(n, 0) = 0 if n > 0
S(n, 1) = S(n, n) = 1
S(n, k) = S(n-1, k-1) + kS(n-1, k)``````

## subsets(\@data[, \$k])

This subroutine iterates over the subsets of data, which is assumed to represent a set. If you pass the optional parameter `\$k` the iteration runs over subsets of data of size `\$k`.

The number of subsets of a set of `n` elements is

``  2**n``

See some values at http://www.research.att.com/~njas/sequences/A000079.

# CORNER CASES

Since version 0.05 subroutines are more forgiving for unsual values of `\$k`:

• If `\$k` is less than zero no tuple exists. Thus, the very first call to the iterator's `next()` method returns `undef`, and a call in list context returns the empty list. (See "DIAGNOSTICS".)

• If `\$k` is zero we have one tuple, the empty tuple. This is a different case than the former: when `\$k` is negative there are no tuples at all, when `\$k` is zero there is one tuple. The rationale for this behaviour is the same rationale for n choose 0 = 1: the empty tuple is a subset of `@data` with `\$k = 0` elements, so it complies with the definition.

• If `\$k` is greater than the size of `@data`, and we are calling a subroutine that does not generate tuples with repetitions, no tuple exists. Thus, the very first call to the iterator's `next()` method returns `undef`, and a call in list context returns the empty list. (See "DIAGNOSTICS".)

In addition, since 0.05 empty `@data`s are supported as well.

# EXPORT

Algorithm::Combinatorics exports nothing by default. Each of the subroutines can be exported on demand, as in

``    use Algorithm::Combinatorics qw(combinations);``

and the tag `all` exports them all:

``    use Algorithm::Combinatorics qw(:all);``

# DIAGNOSTICS

## Warnings

The following warnings may be issued:

Useless use of %s in void context

A subroutine was called in void context.

Parameter k is negative

A subroutine was called with a negative k.

Parameter k is greater than the size of data

A subroutine that does not generate tuples with repetitions was called with a k greater than the size of data.

## Errors

The following errors may be thrown:

Missing parameter data

A subroutine was called with no parameters.

Missing parameter k

A subroutine that requires a second parameter k was called without one.

Parameter data is not an arrayref

The first parameter is not an arrayref (tested with "reftype()" from Scalar::Util.)

# DEPENDENCIES

Algorithm::Combinatorics is known to run under perl 5.6.2. The distribution uses Test::More and FindBin for testing, Scalar::Util for `reftype()`, and XSLoader for XS.

# BUGS

Please report any bugs or feature requests to `bug-algorithm-combinatorics@rt.cpan.org`, or through the web interface at http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Algorithm-Combinatorics.

Math::Combinatorics is a pure Perl module that offers similar features.

List::PowerSet offers a fast pure-Perl generator of power sets that Algorithm::Combinatorics copies and translates to XS.

# BENCHMARKS

There are some benchmarks in the benchmarks directory of the distribution.

 Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations. Addison Wesley Professional, 2005. ISBN 0201853930.

 Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions. Addison Wesley Professional, 2005. ISBN 0201853949.

 Michael Orlov, Efficient Generation of Set Partitions, http://www.informatik.uni-ulm.de/ni/Lehre/WS03/DMM/Software/partitions.pdf.

# AUTHOR

Xavier Noria (FXN), <fxn@cpan.org>