#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11114280
use warnings;
use List::Util qw( uniq );

my @cache = [];

for my $n ( 3 .. 4 )
  {
  print "$_\n" for sort @{ find($n) };
  print "\n";
  }

sub find
  {
  my $n = shift;
  $cache[$n] //= [ uniq $n, map { my $nn = $_;
      map { my $t = $_; map "$t+$_", @{ find($nn) } } @{ find($n - $nn
+) }
      } 1 .. $n - 1 ];
  }
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Outputs:

1+1+1
1+2
2+1
3

1+1+1+1
1+1+2
1+2+1
1+3
2+1+1
2+2
3+1
4
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