I don't like that there are still wrong moves possible*, probably because I din't follow my own concept...

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


my $goal = [3,1];
my ($nl,$nr) = @$goal;

my @results;

pathfinder( [0,0] );                      # start

pp \@$_ for @results;


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

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

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

    }


    # left
    pathfinder( [$l+1,$r  ], @_ ) if $l != $nl;

    # right
    pathfinder( [$l+1,$r+1], @_ ) if $nr != $r;
}
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[[0, 0], [1, 0], [2, 0], [3, 1]]
[[0, 0], [1, 0], [2, 1], [3, 1]]
[[0, 0], [1, 1], [2, 1], [3, 1]]
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*) i.e. the else branch with the warning should never be reached.

update

solution here

Well, for any given target node you build a string of the correct number of left and right moves (or zeroes and ones) and then any permutation is one of the admissible solutions (and all of them).

UPDATE: Just as illustration, using a module, and having to filter out duplicates in the permutations, it could work like this:

use strict;
use warnings;
use Algorithm::Permute;

my $node = '5-2';
my ($all, $right) = split /-/, $node;
my @path = ((0) x ($all-$right),(1) x $right);
my %pathes;
Algorithm::Permute::permute { 
    my $key = join '', @path;
    if( not exists $pathes{$key} ) {
        $pathes{$key} = 1;
        my ($l, $r) = (0,0);
        my $path = "($l-$r) ".join( " ", map{ "(".(++$l)."-".($r+=$_).
+")" } @path);
        print "$path\n";
    }
} @path;
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