# fxn/algorithm-combinatorics

Efficient generation of combinatorial sequences
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```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

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.

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

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

[3] Michael Orlov, *Efficient Generation of Set Partitions*,
http://www.informatik.uni-ulm.de/ni/Lehre/WS03/DMM/Software/partitions.p
df.

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