NAME
Math::Combinatorics  Perform combinations and permutations on lists
SYNOPSIS
Available as an object oriented API.
use Math::Combinatorics;
my @n = qw(a b c);
my $combinat = Math::Combinatorics>new(count => 2,
data => [@n],
);
print "combinations of 2 from: ".join(" ",@n)."\n";
print "".("" x scalar(@n))."\n";
while(my @combo = $combinat>next_combination){
print join(' ', @combo)."\n";
}
print "\n";
print "permutations of 3 from: ".join(" ",@n)."\n";
print "".("" x scalar(@n))."\n";
while(my @permu = $combinat>next_permutation){
print join(' ', @permu)."\n";
}
output:
Or available via exported functions 'permute', 'combine', and 'factorial'.
use Math::Combinatorics;
my @n = qw(a b c);
print "combinations of 2 from: ".join(" ",@n)."\n";
print "".("" x scalar(@n))."\n";
print join("\n", map { join " ", @$_ } combine(2,@n)),"\n";
print "\n";
print "permutations of 3 from: ".join(" ",@n)."\n";
print "".("" x scalar(@n))."\n";
print join("\n", map { join " ", @$_ } permute(@n)),"\n";
Output:
combinations of 2 from: a b c

a b
a c
b c
permutations of 3 from: a b c

a b c
a c b
b a c
b c a
c a b
c b a
Output from both types of calls is the same, but the objectoriented approach consumes much less memory for large sets.
DESCRIPTION
Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties. As a jumping off point, refer to:
http://mathworld.wolfram.com/Combinatorics.html
This module provides a pureperl implementation of nCk, nCRk, nPk, nPRk, !n and n! (combination, multiset, permutation, string, derangement, and factorial, respectively). Functional and objectoriented usages allow problems such as the following to be solved:
 combine  nCk

http://mathworld.wolfram.com/Combination.html
"Fun questions to ask the pizza parlor wait staff: how many possible combinations of 2 toppings can I get on my pizza?".
 derange  !n

http://mathworld.wolfram.com/Derangement.html
"A derangement of n ordered objects, denoted !n, is a permutation in which none of the objects appear in their "natural" (i.e., ordered) place."
 permute  nPk

http://mathworld.wolfram.com/Permutation.html
"Master Mind Game: ways to arrange pieces of different colors in a certain number of positions, without repetition of a color".
Objectoriented usage additionally allows solving these problems by calling "new()" with a frequency vector:
 string  nPRk

http://mathworld.wolfram.com/String.html
"Morse signals: diferent signals of 3 positions using the two symbols  and .".
$o = Math::Combinatorics>new( count=>3 , data=>[qw(. )] , frequency=>[3,3] ); while ( my @x = $o>next_multiset ) { my $p = Math::Combinatorics>new( data=>\@x , frequency=>[map{1} @x] ); while ( my @y = $p>next_string ) { #do something } }
 multiset/multichoose  nCRk

http://mathworld.wolfram.com/Multiset.html
"ways to extract 3 balls at once of a bag with 3 black and 3 white balls".
$o = Math::Combinatorics>new( count=>3 , data=>[qw(white black)] , frequency=>[3,3] ); while ( my @x = $o>next_multiset ) { #do something }
EXPORT
the following export tags will bring a single method into the caller's namespace. no symbols are exported by default. see pod documentation below for method descriptions.
combine
derange
multiset
permute
string
factorial
AUTHOR
Allen Day <allenday@ucla.edu>, with algorithmic contributions from Christopher Eltschka and Tye.
Copyright (c) 20042005 Allen Day. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
ACKNOWLEDGEMENTS
A sincere thanks to everyone for helping to make this a better module. After initial development I've only had time to accept patches and improvements. Math::Combinatorics continues to be developed and improved by the community. Contributors of note include:
For adding new features: Carlos Rica, David Coppit, Carlos Segre, Lyon Lemmens
For bug reports: Ying Yang, Joerg Beyer, Marc Logghe, Yunheng Wang, Torsten Seemann, Gerrit Haase, Joern Behre, Lyon Lemmens, Federico Lucifredi
BUGS / TODO
Report them to the author.
* Need more extensive unit tests.
* tests for new()'s frequency argment
* A known bug (more of a missing feature, actually) does not allow parameterization of k
for nPk in permute(). it is assumed k == n. L</permute()> for details. You can work
around this by making calls to both L</permute()> and L</combine()>
* Lots of really interesting stuff from Mathworld.Wolfram.com. MathWorld rocks! Expect
to see implementation of more concepts from their site, e.g.:
http://mathworld.wolfram.com/BellNumber.html
http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
http://mathworld.wolfram.com/Word.html
* Other combinatorics stuff
http://en.wikipedia.org/wiki/Catalan_number
http://en.wikipedia.org/wiki/Stirling_number
SEE ALSO
String::Combination (alas misnamed, it actually returns permutations on a string).
http://perlmonks.thepen.com/29374.html
http://groups.google.com/groups?selm=38568F79.13680B86%40physik.tumuenchen.de&output=gplain
EXPORTED FUNCTIONS
combine()
Usage : my @combinations = combine($k,@n);
Function: implements nCk (n choose k), or n!/(k!*(nk!)).
returns all unique unorderd combinations of k items from set n.
items in n are assumed to be character data, and are
copied into the return data structure (see "Returns" below).
Example : my @n = qw(a b c);
my @c = combine(2,@n);
print join "\n", map { join " ", @$_ } @c;
# prints:
# b c
# a c
# a b
Returns : a list of arrays, where each array contains a unique combination
of k items from n
Args : a list of items to be combined
Notes : data is internally assumed to be alphanumeric. this is necessary
to efficiently generate combinations of large sets. if you need
combinations of nonalphanumeric data, or on data
C<sort {$a cmp $b}> would not be appropriate, use the
objectoriented API. See L</new()> and the B<compare> option.
Identical items are assumed to be nonunique. That is, calling
C<combine(1,'a','a') yields two sets: {a}, and {a}. See
L</next_multiset() if this is not the desired behavior.
derange()
Usage : my @deranges = derange(@n);
Function: implements !n, a derangement of n items in which none of the
items appear in their originally ordered place.
Example : my @n = qw(a b c);
my @d = derange(@n);
print join "\n", map { join " ", @$_ } @d;
# prints:
# a c b
# b a c
# b c a
# c a b
# c b a
Returns : a list of arrays, where each array contains a derangement of
k items from n (where k == n).
Args : a list of items to be deranged.
Note : k should really be parameterizable. this will happen
in a later version of the module. send me a patch to
make that version come out sooner.
Notes : data is internally assumed to be alphanumeric. this is necessary
to efficiently generate combinations of large sets. if you need
combinations of nonalphanumeric data, or on data
C<sort {$a cmp $b}> would not be appropriate, use the
objectoriented API. See L</new()>, and the B<compare> option.
next_derangement()
Usage : my @derangement = $c>next_derangement();
Function: get derangements for @data.
Returns : returns a permutation of items from @data (see L</new()>),
where none of the items appear in their natural order. repeated calls
retrieve all unique derangements of @data elements. a returned empty
list signifies all derangements have been iterated.
Args : none.
factorial()
Usage : my $f = factorial(4); #returns 24, or 4*3*2*1
Function: calculates n! (n factorial).
Returns : undef if n is noninteger or n < 0
Args : a positive, nonzero integer
Note : this function is used internally by combine() and permute()
permute()
Usage : my @permutations = permute(@n);
Function: implements nPk (n permute k) (where k == n), or n!/(nk)!
returns all unique permutations of k items from set n
(where n == k, see "Note" below). items in n are assumed to
be character data, and are copied into the return data
structure.
Example : my @n = qw(a b c);
my @p = permute(@n);
print join "\n", map { join " ", @$_ } @p;
# prints:
# b a c
# b c a
# c b a
# c a b
# a c b
# a b c
Returns : a list of arrays, where each array contains a permutation of
k items from n (where k == n).
Args : a list of items to be permuted.
Note : k should really be parameterizable. this will happen
in a later version of the module. send me a patch to
make that version come out sooner.
Notes : data is internally assumed to be alphanumeric. this is necessary
to efficiently generate combinations of large sets. if you need
combinations of nonalphanumeric data, or on data
C<sort {$a cmp $b}> would not be appropriate, use the
objectoriented API. See L</new()>, and the B<compare> option.
Identical items are assumed to be nonunique. That is, calling
C<permute('a','a') yields two sets: {a,a}, and {a,a}. See
L</next_string() if this is not the desired behavior.
CONSTRUCTOR
new()
Usage : my $c = Math::Combinatorics>new( count => 2, #treated as int
data => [1,2,3,4] #arrayref or anonymous array
);
Function: build a new Math::Combinatorics object.
Returns : a Math::Combinatorics object
Args : count  required for combinatoric functions/methods. number of elements to be
present in returned set(s).
data  required for combinatoric B<AND> permutagenic functions/methods. this is the
set elements are chosen from. B<NOTE>: this array is modified in place; make
a copy of your array if the order matters in the caller's space.
frequency  optional vector of data frequencies. must be the same length as the B<data>
constructor argument. These two constructor calls here are equivalent:
$a = 'a';
$b = 'b';
Math::Combinatorics>new( count=>2, data=>[\$a,\$a,\$a,\$a,\$a,\$b,\$b] );
Math::Combinatorics>new( count=>2, data=>[\$a,\$b], frequency=>[5,2] );
so why use this? sometimes it's useful to have multiple identical entities in
a set (in set theory jargon, this is called a "bag", See L<Set::Bag>).
compare  optional subroutine reference used in sorting elements of the set. examples:
#appropriate for character elements
compare => sub { $_[0] cmp $_[1] }
#appropriate for numeric elements
compare => sub { $_[0] <=> $_[1] }
#appropriate for object elements, perhaps
compare => sub { $_[0]>value <=> $_[1]>value }
The default sort mechanism is based on references, and cannot be predicted.
Improvements for a more flexible compare() mechanism are most welcome.
OBJECT METHODS
next_combination()
Usage : my @combo = $c>next_combination();
Function: get combinations of size $count from @data.
Returns : returns a combination of $count items from @data (see L</new()>).
repeated calls retrieve all unique combinations of $count elements.
a returned empty list signifies all combinations have been iterated.
Note : this method may only be used if a B<frequency> argument is B<NOT>
given to L</new()>, otherwise use L</next_multiset()>.
Args : none.
next_multiset()
Usage : my @multiset = $c>next_multiset();
Function: get multisets for @data.
Returns : returns a multiset of items from @data (see L</new()>).
a multiset is a special type of combination where the set from which
combinations are drawn contains items that are indistinguishable. use
L</next_multiset()> when a B<frequency> argument is passed to L</new()>.
repeated calls retrieve all unique multisets of @data elements. a
returned empty list signifies all multisets have been iterated.
Note : this method may only be used if a B<frequency> argument is given to
L</new()>, otherwise use L</next_combination()>.
Args : none.
next_permutation()
Usage : my @permu = $c>next_permutation();
Function: get permutations of elements in @data.
Returns : returns a permutation of items from @data (see L</new()>).
repeated calls retrieve all unique permutations of @data elements.
a returned empty list signifies all permutations have been iterated.
Note : this method may only be used if a B<frequency> argument is B<NOT>
given to L</new()>, otherwise use L</next_string()>.
Args : none.
next_string()
Usage : my @string = $c>next_string();
Function: get strings for @data.
Returns : returns a multiset of items from @data (see L</new()>).
a multiset is a special type of permutation where the set from which
combinations are drawn contains items that are indistinguishable. use
L</next_permutation()> when a B<frequency> argument is passed to L</new()>.
repeated calls retrieve all unique multisets of @data elements. a
returned empty list signifies all strings have been iterated.
Note : this method may only be used if a B<frequency> argument is given to
L</new()>, otherwise use L</next_permutation()>.
Args : none.
INTERNAL FUNCTIONS AND METHODS
sum()
Usage : my $sum = sum(1,2,3); # returns 6
Function: sums a list of integers. noninteger list elements are ignored
Returns : sum of integer items in arguments passed in
Args : a list of integers
Note : this function is used internally by combine()
compare()
Usage : $obj>compare()
Function: internal, undocumented. holds a comparison coderef.
Returns : value of compare (a coderef)
count()
Usage : $obj>count()
Function: internal, undocumented. holds the "k" in nCk or nPk.
Returns : value of count (an int)
data()
Usage : $obj>data()
Function: internal, undocumented. holds the set "n" in nCk or nPk.
Returns : value of data (an arrayref)
swap()
internal, undocumented.
reverse()
internal, undocumented.
rotate()
internal, undocumented.
upper_bound()
internal, undocumented.
lower_bound()
internal, undocumented.
_permutation_cursor()
Usage : $obj>_permutation_cursor()
Function: internal method. cursor on permutation iterator order.
Returns : value of _permutation_cursor (an arrayref)
Args : none