I figure that returning a single variable ($zaza)is more efficient than returning $aob$x$y

Returning $aob[$x][$y], is returning a single variable. Whether you derefence the arrays here:

            $zaza = $aob[$x][$y];
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Or here:

    return $aob[$x][$y];
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Makes no difference.

However, using my for ( $x, $y, $z ) & $zaza would make some difference as lexicals are more efficient than globals. Plus you could then benefit from use strict.

But your subroutine can be refactored as:

sub popnum1 {
    my( $x, $y, $z ) = @_;

    if ( $y == 0 ) {
        return $aob[$x][0] = $initial + $z;
    }
    else {
        if ( substr( $aod[ $y - 1 ], $x, 1 ) ne 'a' ) {
            return $aob[$x][$y] = $initial + $z;

        }
        else {
            return $aob[$x][$y] = $z + $aob[$x][ $y - 1 ];
        }
    }
}
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which saves a temporary variable and two, double dereferences.

Personally, I think I'd code that as:

sub popnum1 {
    my( $x, $y, $z ) = @_;

    return $aob[ $x ][ $y ] = 
        $y && substr( $aod[ $y - 1 ], $x, 1 ) ne 'a'
            ? $initial + $z
            : $z + $aob[$x][ $y - 1 ];
}
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Though I'd want to verify that my logic transformation was correct. That should be appreciably more efficient than your original above.