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P(m,n) = {
   1                    if m or n = 1,
...
   1 + P(m,m-1)         if m = n > 1,
   P(m,n-1) + P(m-n,n)  if m > n > 1
}

sub partition {
    my ( $m, $n ) = @_;
...
    $n ||= $m;
    say partition( $m, $n );
}

sub P {
    # Returns strings!
...
        printf( "\n = %s", $_ );   
    }
}

$ partition.pl 7
P(7,7)
...
 = 7 + 1 + P(3,1) + P(1,2) + 1 + 1 + P(2,1) + 1 + 1
 = 12 + 1 + 1 + 1
 = 15