All computations are in the S_n-invariant part.

When there are two lines, relation rank is first, Gorenstein rank second.
When there is just one line, the ranks are known to be equal.

---------------------------------
n=0:
  g=2:  1   1
  g=3:  1   2   2   1
  g=4:  1   3   6   6   3
  g=5:  1   3  10  19  19  10   3
  g=6:  1   4  15  42  71  72
        1              71  71
  g=7:  1   4  20  69 171   ?   ?
        1
---------------------------------
n=1:
  g=2:  1   2   1
  g=3:  1   4   7   4   1
  g=4:  1   5  17  25  17
  g=5:  1   6  28  75 107  75
        1             107
  g=6:  1   7  41 152   ?   ?   ?
        1
---------------------------------
n=2:
  g=2:  1   4   4   1
  g=3:  1   6  17  17   6
  g=4:  1   8  36  81  81
  g=5:  1   9  57 205 405 406
        1
---------------------------------
n=3:
  g=2:  1   5  10   5
  g=3:  1   8  32  52  32
  g=4:  1  10  62 190 285 190
        1             285
  g=5:  1  12  95 441   ?   ?   ?
        1
---------------------------------
n=4:
  g=2:  1   7  20  20
  g=3:  1  10  53 125 125
  g=4:  1  13  96 387 799 799
        1
---------------------------------
n=5:
  g=2:  1   8  33  51  33
        1          51
  g=3:  1  12  78 245 369   ?
        1
  g=4:  1  15 137 681   ?   ?   ?
        1
---------------------------------