Allen Day

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 object-oriented 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 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
 }

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) 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.

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

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

EXPORTED FUNCTIONS

combine()

 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.

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

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 non-integer or n < 0
 Args    : a positive, non-zero 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!/(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.

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.  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()

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



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