Math::Combinatorics - Perform combinations and permutations on lists

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 object-oriented approach consumes much less memory for large sets.

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 pure-perl implementation of nCk, nCRk, nPk, nPRk, !n and n! (combination, multiset, permutation, string, derangement, and factorial, respectively). Functional and object-oriented 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.

Object-oriented 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
 }

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

Allen Day <allenday@ucla.edu>, with algorithmic contributions from Christopher Eltschka and Tye.

Copyright (c) 2004-2005 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.

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

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

Set::Scalar

Set::Bag

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.tu-muenchen.de&output=gplain

 Usage   : my @combinations = combine($k,@n);
 Function: implements nCk (n choose k), or n!/(k!*(n-k!)).
           returns all unique unorderd combinations of k items from 
           set n.  items in n are assumed to be character data, and 
           are copied into the return data structure (see "Returns" 
           below).
 Example : my @n = qw(a b c);
           my @c = combine(2,@n);
           print join "\n", map { join " ", @$_ } @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 non-alphanumeric data, 
           or on data C<sort {$a cmp $b}> would not be appropriate, 
           use the object-oriented API.  See L</new()> and the 
           B<compare> option.

           Identical items are assumed to be non-unique.  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.

 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 " ", @$_ } @d;
           # prints:
           # a c b
           # b a c
           # b c a
           # c a b
           # c b a
 Returns : a list of arrays, where each array contains a derangement of
           k items from n (where k == n).
 Args    : a list of items to be deranged.
 Note    : k should really be parameterizable.  this will happen
           in a later version of the module.  send me a patch to
           make that version come out sooner.
 Notes   : data is internally assumed to be alphanumeric.  this is 
           necessary to efficiently generate combinations of large sets.  
           if you need combinations of non-alphanumeric data, or on data
           C<sort {$a cmp $b}> would not be appropriate, use the
           object-oriented API.  See L</new()>, and the B<compare> 
           option.

 Usage   : my $f = factorial(4); #returns 24, or 4*3*2*1
 Function: calculates n! (n factorial).
 Returns : undef if n is non-integer or n < 0
 Args    : a positive, non-zero integer
 Note    : this function is used internally by combine() and permute()

 Usage   : my @permutations = permute(@n);
 Function: implements nPk (n permute k) (where k == n), or n!/(n-k)!
            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 " ", @$_ } @p;
           # prints:
           # b a c
           # b c a
           # c b a
           # c a b
           # a c b
           # a b c
 Returns : a list of arrays, where each array contains a permutation of
           k items from n (where k == n).
 Args    : a list of items to be permuted.
 Note    : k should really be parameterizable.  this will happen
           in a later version of the module.  send me a patch to
           make that version come out sooner.
 Notes   : data is internally assumed to be alphanumeric.  this is necessary
           to efficiently generate combinations of large sets.  if you need
           combinations of non-alphanumeric data, or on data
           C<sort {$a cmp $b}> would not be appropriate, use the
           object-oriented API.  See L</new()>, and the B<compare> option.

           Identical items are assumed to be non-unique.  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.

 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.

 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.

 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.

 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.

 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.

 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.

 Usage   : my $sum = sum(1,2,3); # returns 6
 Function: sums a list of integers.  non-integer 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()

 Usage   : $obj->compare()
 Function: internal, undocumented.  holds a comparison coderef.
 Returns : value of compare (a coderef)

 Usage   : $obj->count()
 Function: internal, undocumented.  holds the "k" in nCk or nPk.
 Returns : value of count (an int)

 Usage   : $obj->data()
 Function: internal, undocumented.  holds the set "n" in nCk or nPk.
 Returns : value of data (an arrayref)

internal, undocumented.

internal, undocumented.

internal, undocumented.

internal, undocumented.

internal, undocumented.

 Usage   : $obj->_permutation_cursor()
 Function: internal method.  cursor on permutation iterator order.
 Returns : value of _permutation_cursor (an arrayref)
 Args    : none