n=1, k>=2:
dim S_k  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 4, 0, 3, 0]
k                                      12

n=2, k>=3:
dim S_k  [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0, 7, 0, 8, 0, 9, 0, 11, 1, 13, 0, 13, 1, 17, 1, 18, 1, 20, 2, 23, 3, 26, 2, 27, 4]
new      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0,  6, 1,  8, 0,  8, 1, 11, 1, 12, 1, 14, 2, 16, 3, 19, 2, 20, 4]
k                             10    12    14

n=3, k>=4
dim S_k  [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 4, 0, 6, 0, 9, 0, 14, 0, 17, 0, 27, 0, 34, 0, 46, 0, 61, 0, 82, 0, 99, 0, 135, 0, 165, 0, 208, 0, 261, 0, 325, 0, 389, 0]
new      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0,  5, 0,  7, 0, 15, 0, 18, 0, 28, 0, 41, 0, 57, 0, 72, 0, 105, 0, 129, 0, 169, 0, 219, 0, 276, 0, 337, 0]
k                                12    14    16

n=4, k>=5:
dim S_k  [0, 0, 0, 1, 0, 1, 0, 2, 0, 3, 0, 7, 0, 12, 1, 22, 1, 38, 3, 68, 6, 110, 15, 193, 29, 315, 68, 520, 135, 840, 262, 1350, 500, 2116, 917, 3321, 1597, 5092, 2763, 7762, 4588, 11656, 7418, 17300, 11788, 25393, 18333]
new      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1, 0,  5, 0, 12, 0, 28, 2,  59, 10, 119, 21, 217, 57, 393, 122, 677, 243, 1138, 477, 1861, 889, 3001, 1560, 4702, 2718, 7291, 4536, 11096, 7351, 16623, 11709, 24610, 18242]
k                  8    10    12    14    16

n=5, k>=6:
dim S_k  [0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 13, 0, 28, 0, 76, 0, 186, 0, 491, 0, 1202, 0, 3083, 0, 7608, 0, 18611, 0, 44155, 0, 102364, 0, 229722, 0, 499896, 0, 1056147, 0, 2165577, 0, 4313086, 0, 8366754, 0, 15820348, 0]
new      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1, 0,  2, 0, 26, 0,  91, 0, 306, 0,  904, 0, 2583, 0, 6796, 0, 17377, 0, 42323, 0,  99589, 0, 225788, 0, 494353, 0, 1048326, 0, 2154862, 0, 4298784, 0, 8347299, 0, 15794692, 0]
k                          12    14

n=6, k>=7:
dim S_k  [0, 0, 0, 1, 0, 3, 0, 9, 0, 33, 0, 117, 1, 486]
new      [0, 0, 0, 0, 0, 0, 0, 0, 0,  2, 0,  28, 0, 238]
k                 10    12

n=7, k>=8:
dim S_k  [0, 0, 0, 0, 3, 0, 9, 0, 83]
new      [0, 0, 0, 0, 0, 0, 0, 0, 12]
k                    12    14     16

n=8, k>=9:
dim S_k (lower bound)  [0, 1, 0, 4, 1, 23, 2, 234]
k                         10

n=9, k>=10:
dim S_k (lower bound)  [0, 0, 2, 0, 25, 0, 843]
k                            12

n=10, k>=11:
dim S_k (lower bound)  [0, 2, 0, 43, 1, 1591]
k                         12

n=11, k>=12:
dim S_k (lower bound)  [1, 0, 32, 0, 6478]
k                      12