The confusing "problem" is the coordinate system, it's trivial if you change it to left-down and right-down moves

               0-0
            1-0   0-1
         2-0   1-1   0-2
      3-0  *2-1   1-2   0-3
   4-0   3-1   2-2   1-3   0-4
5-0   4-1   3-2   2-3   1-4   0-5

[download]

You see in your system:

So your goal needs 2 left and 1 right move in any order, because this is just a distorted rectangle with 2 down and 1 right move

0-0   0-1   0-2   0-3   0-4   0-5
1-0   1-1   1-2   1-3   1-4
2-0  *2-1   2-2   2-3
3-0   3-1   3-2
4-0   4-1
5-0

[download]

This algorithm is fixing it by recalculating the remaining left-down moves

use strict;
use warnings;
use Data::Dump qw/pp dd/;


my $goal = [3,1];
my ($gl,$gr) = @$goal;

my @results;

pathfinder( [0,0,"start"] );                      # start

pp \@$_ for @results;


sub pathfinder {
    my ( $last ) = @_;

    my ( $l, $r ) = @$last ;

    if ( $gl == $l ) {
        if ($gr == $r) {
            push @results,[ reverse @_];
        } else {
            warn "wrong",pp [reverse @_];
            return
        }

    }


    # left
    pathfinder( [$l+1,$r  ,"left"], @_ ) if $l < $gl - ($gr - $r);

    # right
    pathfinder( [$l+1,$r+1,"right"], @_ ) if $r < $gr;
}

[download]

C:/Perl_64/bin\perl.exe  d:/tmp/pascale_path.pl 
[
  [0, 0, "start"],
  [1, 0, "left"],
  [2, 0, "left"],
  [3, 1, "right"],
]
[
  [0, 0, "start"],
  [1, 0, "left"],
  [2, 1, "right"],
  [3, 1, "left"],
]
[
  [0, 0, "start"],
  [1, 1, "right"],
  [2, 1, "left"],
  [3, 1, "left"],
]

Compilation finished at Thu Mar 22 12:41:53

[download]

update

in hindsight it's probably better implemented in the new coordinate system and only the results are transformed back.

The code is much easier then.

Cheers Rolf
(addicted to the Perl Programming Language and ☆☆☆☆ :)
Wikisyntax for the Monastery

In reply to Re: compute paths in Pascal's triangle (aka Tartaglia's one) (2 updates) by LanX
in thread compute paths in Pascal's triangle (aka Tartaglia's one) by Discipulus

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