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Re: x objects in y containers where all objects are used (multinomial coefficients)

by repellent (Priest)
on Nov 09, 2009 at 20:42 UTC ( #806018=note: print w/ replies, xml ) Need Help??


in reply to x objects in y containers where all objects are used

This is a multinomial coefficients problem. The assumptions are:

  • no empty containers
  • order of items in containers do not matter in the results
  • order of containers do matter in the results

mult_coeff(3, qw(a b c d e)); yields 150 results, which corresponds to the math:
5! 5! ------------ * 3 + ------------ * 3 = 150 1! * 2! * 2! 1! * 1! * 3!

for combinations of 1-2-2, 2-1-2, 2-2-1, 1-1-3, 1-3-1, 3-1-1.

nck_with_leftover() should be memoized for performance.

A solution:
sub mult_coeff { my $c = shift(); return [] if $c <= 0; return [ [ @_ ] ] if $c == 1; my @sets; for my $k (1 .. @_ - $c + 1) { for my $nck_ref (nck_with_leftover($k, @_)) { push @sets, map { unshift(@{ $_ }, [ @{ $nck_ref->[0] } ]); # clone r +eference $_ } mult_coeff($c - 1, @{ $nck_ref->[1] }); } } return @sets; } sub nck_with_leftover { my $k = shift(); return [ [], [ @_ ] ] if $k <= 0; my @groups; my @leftover; while (@_) { my $pick = shift(); push @groups, map { unshift(@{ $_->[0] }, $pick); unshift(@{ $_->[1] }, @leftover); $_ } nck_with_leftover($k - 1, @_); push @leftover, $pick; } return @groups; } use Data::Dumper; my @results = mult_coeff(3, qw(a b c d e)); print Dumper \@results;


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