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### Math::Combinatorics

by Lexicon (Chaplain)
 on Apr 24, 2001 at 13:48 UTC Need Help??
 Category: Mathematics Author/Contact Info Alex Scouras lexicon@anapraxis.net http://code.anapraxis.net Description: This is a module which impliments the Combinatorics functions Pick and Choose. POD for this module is located at http://code.anapraxis.net. Questions and Comments are encouraged, as I intend to release this to the Perl community via CPAN. In addition to normal functionality, Pick and Choose have flags to calculate with repetition and/or to auto-sum across R for an assumed set of R-Combinations/R-Permutations where 0 <= R <= N. ``` #!/usr/local/bin/perl -w package Math::Combinatorics; use strict; use Exporter; @Math::Combinatorics::ISA = qw(Exporter); @Math::Combinatorics::EXPORT = ""; @Math::Combinatorics::EXPORT_OK = qw( Pick Choose Permutation Combination ); %Math::Combinatorics::EXPORT_TAGS = ( "common" => [qw(Pick Choose )], "formal" => [qw(Permutation Combination)], ); \$Math::Combinatorics::VERSION = 0.91; # v1.00 Release Candidate. #################################################################### # CHOOSE ( # N, # Size of Master Array # R, # Size of Subset # F # Repetition Flag # ) #################################################################### # Of note, 1) (N-R)! divides out a large portion of N!. # 2) Choose(N,R) = Choose(N, N-R). # 3) Choose(N,R,'r') = Choose(N+R-1,R). # # These facts are taken advantage of in the formula to increase # speed and improve accuracy. # # In the special case of R = -1, the sum of R-Combinations for all # R = 0..N is returned. This happens to be simply 2^n. #################################################################### sub Choose { my \$n = shift; die "N (\$n) must be a positive integer" if \$n < 1; my \$r = shift; my \$f = shift || ''; # SUMMATION ACROSS R --------------------------------------------- if (\$r eq '*') { # NO REPETITION if (!\$f) { return 1 << \$n # REPETITION } elsif (\$f eq 'r') { my \$sum = 0; for my \$r (0..\$n) { my \$c = 1; for (1..\$r) { \$c *= \$n + \$_ - 1; # n! / (n-r)! \$c /= \$_; # c / r! } \$sum += \$c; } return \$sum; # INVALID FLAG } else { die "Invalid Flag: '\$f'" } # SPECIFIED R ---------------------------------------------------- } else { # NO REPETITION if (!\$f) { die "R must be 0 < R < N (\$n) if there is no repetition" if (\$r < 0 || \$r > \$n); my \$c = 1; if (\$r > \$n/2) { \$r = \$n - \$r } # Take advantage of 2) for (1..\$r) { \$c *= \$n--; # n! / (n-r)! \$c /= \$_; # c / r! } return \$c; # REPETITION } elsif (\$f eq 'r') { die "R (\$r) must be 0 < R if there is repetition" if (\$r < 0); \$n += \$r - 1; my \$c = 1; if (\$r > \$n/2) { \$r = \$n - \$r } # Take advantage of 2) for (1..\$r) { \$c *= \$n--; # n! / (n-r)! \$c /= \$_; # c / r! } return \$c; # INVALID FLAG } else { die "Invalid Flag: '\$f'" } } } #################################################################### # PICK ( # N, # Size of Master Array # R, # Length of Sub-Sequence # F # Repetition Flag # ) #################################################################### sub Pick { my \$n = shift; die "N (\$n) must be a positive integer" if \$n < 1; my \$r = shift; my \$f = shift || ''; # SUMMATION ACROSS R --------------------------------------------- if (\$r eq '*') { # NO REPETITION if (!\$f) { my \$sum = 0; for my \$r (0..\$n) { my \$p = 1; \$p *= \$_ for (\$n-\$r+1..\$n); # n! / (n-r)! \$sum += \$p; } return \$sum; # REPETITION } elsif (\$f eq 'r') { my \$sum = 0; for my \$r (0..\$n) { \$sum += \$n ** \$r; } return \$sum; # INVALID FLAG } else { die "Invalid Flag: '\$f'" } # SPECIFIED R ---------------------------------------------------- } else { # NO REPETITION if (!\$f) { my \$p = 1; \$p *= \$_ for (\$n-\$r+1..\$n); # n! / (n-r)! return \$p; # REPETITION } elsif (\$f eq 'r') { return \$n ** \$r; # INVALID FLAG } else { die "Invalid Flag: '\$f'" } } } *Combination = *Choose; *Permutation = *Pick; 1; =pod =head1 NAME Math::Combinatorics - Pick and Choose combinatorics functions =head1 SYNOPSIS use Math::Combinatorics; use strict; my @master = ( 1, 2, 3, 4, 5 ); # The master array my \$n = @master; # Size of the master my \$r = rand * \$n # Size of the subset \$Combinations = Choose(\$n, \$r) # Specific R, No Repetition. # C = N! / ((N-R)! * R!) # O(\$r) \$Combinations = Choose(\$n, -1) # Sum of R's, No Repetition # C = 2^N # O(1) \$Combinations = Choose(\$n, \$r, 'r') # Specific R, Repetition # C = (N+R-1)! / ((N-1)! * R!) # O(\$r) \$Combinations = Choose(\$n, -1, 'r') # Sum of R's, Repetition. # C = Sum of Chooses over R # O(\$r~2/2 + \$r/2) \$Combinations = Combination(\$n, \$r) # &Combination == &Choose \$Permutations = Pick(\$n, \$r) # Specific R, No Repetition. # P = N! / (N-R)! # O(\$r^2/2 + \$r/2) \$Permutations = Pick(\$n, -1) # Sum of R's, No Repetition # P = Sum of Picks over R. # O(1) \$Permutations = Pick(\$n, \$r, 'r') # Specific R, Repetition # P = N^R # O(1) \$Permutations = Pick(\$n, -1, 'r') # Sum of R's, Repetition. # P = Sum of Pics over R. # O(\$r) \$Permutations = Permutation(\$n, \$r) # &Permutation == &Pick =head1 DESCRIPTION This module only includes two functions, Pick and Choose. Pick returns R-Permutations of a set, that is, sub-sequences of a set. Think, "How many words can I make with my tiles in Scrabble?" Lots, you rearrange the letters and get new words. For formality, &Pick has been aliased to &Permutation. You may use them interchangably. Choose returns R-Combinations, that is, subsets. Order is irrelevant. Think, "How many hands can I make with my cards in 5-Card Stud Poker?". One, your hand is the same no matter what order you put the cards in. For formality, Choose has been aliased to &Combination. You may use them interchangably. Each function has 2 flags, leading to 4 execution modes each, for a total of 8 different 'functions' that the module can currently perform. Normally a function executes without allowing repetitions, but by setting \$f to 'r' repetitions will be permitted. \$f is optional, and will default to False ( no repetition ). The other is more of a pseudo-flag, \$r. Normally \$r should be 0 <= \$r <= \$n. However, as a special case, if \$r is set to '*', the summation of all combinations (or permutations) for all \$r = 1..\$n will be performed, returning the total count of subsets (or sub-sequences) of a Master array. =head1 BUGS & WARNINGS This seems a rather tiny file to release as module, and could probably do with some expansion. If you have anything you would like to see in this module or, of course, find an error, let me know. If you have code you think belongs here, let me know and I'll toss it in and put you in the credits. This module doesn't export anything by default. You should pick your two combinatorics functions by hand by calling use with the 'formal' or 'common' tags. use Math::Combinatorics qw(:common); # &Pick & &Choose use Math::Combinatorics qw(:formal); # &Permutation & &Combiantion =head1 VERSION 25 April 2001 - Version 0.91 (1.0 Release Candidate B) =over4 =item * Provided aliases of Combination for Choose and Permutation for Pick. =item * Reduced lines to 70 characters from 71. =back 22 April 2001 - Version 0.91 (1.0 Release Candidate A) =over4 =item * Original Public Beta =back =head1 CREDITS Copyright (c) 2001 - Alexander (Lexicon) Scouras - Anapraxis.Net All rights reserved. For current release information see CPAN or F. Bug reports or comments may be sent to F This program is free software. It may be distributed and/or modified under either the Perl Artistic License or the GNU General Public License. =cut ```

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