complex cdim f-vector faces facets pure matroid graph pdim codim degree m-vector aED pED ML [1,2,3,4,5] 0 (1,5) 6 5 Y Y Y 31 0 1 () 1 1 N/A [12,3,4,5] 1 (1,5,1) 7 4 N N N 30 1 2 (1) 5 2 N/A [12,13,4,5] 1 (1,5,2) 8 4 N N N 29 2 3 (3) 5 2 N/A [12,13,14,5] 1 (1,5,3) 9 4 N N N 28 3 4 (6) 5 2 N/A [12,13,14,15] 1 (1,5,4) 10 4 Y Y Y 27 4 5 (10) 5 2 N/A [12,13,23,4,5] 1 (1,5,3) 9 5 N N N 28 3 5 (5) 21 19 N/A [15,23,4] 1 (1,5,2) 8 3 N N N 29 2 4 (2) 25 25 N/A [13,15,23,4] 1 (1,5,3) 9 4 N N N 28 3 5 (5) 21 8 N/A [12,13,15,23,4] 1 (1,5,4) 10 5 N N N 27 4 7 (8) 33 25 N/A [14,15,23] 1 (1,5,3) 9 3 Y N N 28 3 6 (4) 25 25 N/A [12,14,15,23] 1 (1,5,4) 10 4 Y N N 27 4 7 (8) 21 8 N/A [12,13,14,15,23] 1 (1,5,5) 11 5 Y N N 26 5 9 (12) 33 25 N/A [13,14,23,24,5] 1 (1,5,4) 10 5 N N N 27 4 6 (9) 9 3 N/A [12,13,14,23,24,5] 1 (1,5,5) 11 6 N N N 26 5 9 (12) 25 25 N/A [13,15,23,24] 1 (1,5,4) 10 4 Y N N 27 4 8 (7) 53 29 N/A [12,13,15,23,24] 1 (1,5,5) 11 5 Y N N 26 5 10 (11) 61 29 N/A [13,14,15,23,24] 1 (1,5,5) 11 5 Y N N 26 5 9 (12) 25 8 N/A [12,13,14,15,23,24] 1 (1,5,6) 12 6 Y N N 25 6 12 (16) 37 29 N/A [13,14,15,23,24,25] 1 (1,5,6) 12 6 Y Y Y 25 6 10 (18) 9 3 N/A [12,13,14,15,23,24,25] 1 (1,5,7) 13 7 Y Y Y 24 7 14 (22) 25 25 N/A [13,14,25,34] 1 (1,5,4) 10 4 Y N N 27 4 10 (6) 105 143 N/A [12,13,14,25,34] 1 (1,5,5) 11 5 Y N N 26 5 12 (10) 113 88 N/A [12,13,14,23,25,34] 1 (1,5,6) 12 6 Y N N 25 6 14 (15) 101 104 N/A [14,15,23,25,34] 1 (1,5,5) 11 5 Y N N 26 5 12 (10) 113 94 N/A [12,14,15,23,25,34] 1 (1,5,6) 12 6 Y N N 25 6 14 (15) 81 52 N/A [12,13,14,15,25,34] 1 (1,5,6) 12 6 Y N N 25 6 15 (14,1) 137 135 N/A [12,13,14,15,23,25,34] 1 (1,5,7) 13 7 Y N N 24 7 17 (20) 105 103 N/A [12,13,14,23,24,34,5] 1 (1,5,6) 12 7 N N N 25 6 12 (16) 73 85 N/A [12,13,14,23,24,25,34] 1 (1,5,7) 13 7 Y N N 24 7 17 (20) 145 131 N/A [13,14,15,23,24,25,34] 1 (1,5,7) 13 7 Y N N 24 7 16 (21) 49 42 N/A [12,13,14,15,23,24,25,34] 1 (1,5,8) 14 8 Y N N 23 8 20 (26) 129 146 N/A [12,13,14,15,24,25,34,35] 1 (1,5,8) 14 8 Y Y Y 23 8 19 (27) 41 45 N/A [12,13,14,15,23,24,25,34,35] 1 (1,5,9) 15 9 Y Y Y 22 9 23 (33) 89 109 N/A [12,13,14,15,23,24,25,34,35,45] 1 (1,5,10) 16 10 Y Y Y 21 10 27 (40) 233 309 N/A [123,4,5] 2 (1,5,3,1) 10 3 N N N 27 4 6 (9) 17 6 N/A [123,14,5] 2 (1,5,4,1) 11 3 N N N 26 5 8 (13) 21 8 N/A [123,14,15] 2 (1,5,5,1) 12 3 N N N 25 6 10 (18) 21 8 N/A [123,15,24] 2 (1,5,5,1) 12 3 N N N 25 6 11 (17) 25 8 N/A [123,14,24,5] 2 (1,5,5,1) 12 4 N N N 25 6 11 (17) 57 29 N/A [123,14,15,24] 2 (1,5,6,1) 13 4 N N N 24 7 14 (22) 61 29 N/A [123,14,15,24,25] 2 (1,5,7,1) 14 5 N N N 23 8 17 (28) 61 33 N/A [123,15,25,34] 2 (1,5,6,1) 13 4 N N N 24 7 16 (21) 93 50 N/A [123,14,15,25,34] 2 (1,5,7,1) 14 5 N N N 23 8 21 (26) 205 161 N/A [123,14,24,34,5] 2 (1,5,6,1) 13 5 N N N 24 7 16 (21) 117 112 N/A [123,14,24,25,34] 2 (1,5,7,1) 14 5 N N N 23 8 21 (26) 149 136 N/A [123,14,15,24,25,34] 2 (1,5,8,1) 15 6 N N N 22 9 26 (32) 209 241 N/A [123,45] 2 (1,5,4,1) 11 2 N N N 26 5 12 (10) 85 98 N/A [123,14,45] 2 (1,5,5,1) 12 3 N N N 25 6 14 (15) 101 50 N/A [123,14,15,45] 2 (1,5,6,1) 13 4 N N N 24 7 18 (20) 149 107 N/A [123,15,24,45] 2 (1,5,6,1) 13 4 N N N 24 7 16 (21) 77 30 N/A [123,14,24,45] 2 (1,5,6,1) 13 4 N N N 24 7 17 (20) 149 101 N/A [123,14,15,24,45] 2 (1,5,7,1) 14 5 N N N 23 8 21 (26) 161 125 N/A [123,14,24,34,45] 2 (1,5,7,1) 14 5 N N N 23 8 22 (25) 221 167 N/A [123,15,24,34,45] 2 (1,5,7,1) 14 5 N N N 23 8 19 (27) 101 45 N/A [123,14,15,24,34,45] 2 (1,5,8,1) 15 6 N N N 22 9 26 (32) 197 193 N/A [123,14,15,25,34,45] 2 (1,5,8,1) 15 6 N N N 22 9 24 (33) 165 123 N/A [123,14,15,24,25,45] 2 (1,5,8,1) 15 6 N N N 22 9 26 (32) 277 242 N/A [123,14,15,24,25,34,45] 2 (1,5,9,1) 16 7 N N N 21 10 31 (39) 293 326 N/A [123,14,15,24,25,34,35] 2 (1,5,9,1) 16 7 N N N 21 10 31 (39) 141 155 N/A [123,14,15,24,25,34,35,45] 2 (1,5,10,1) 17 8 N N N 20 11 38 (46) 453 505 N/A [123,124,5] 2 (1,5,5,2) 13 3 N N N 24 7 12 (24) 25 8 N/A [123,124,15] 2 (1,5,6,2) 14 3 N N N 23 8 15 (30) 25 8 N/A [123,124,15,25] 2 (1,5,7,2) 15 4 N N N 22 9 19 (36) 65 29 N/A [123,124,34,5] 2 (1,5,6,2) 14 4 N N N 23 8 18 (28) 89 90 N/A [123,124,25,34] 2 (1,5,7,2) 15 4 N N N 22 9 23 (34) 101 104 N/A [123,124,15,25,34] 2 (1,5,8,2) 16 5 N N N 21 10 30 (40) 281 318 N/A [123,124,35] 2 (1,5,6,2) 14 3 N N N 23 8 18 (28) 77 28 N/A [123,124,15,35] 2 (1,5,7,2) 15 4 N N N 22 9 23 (34) 133 67 N/A [123,124,34,35] 2 (1,5,7,2) 15 4 N N N 22 9 24 (33) 153 110 N/A [123,124,15,34,35] 2 (1,5,8,2) 16 5 N N N 21 10 29 (40) 205 179 N/A [123,124,15,25,35] 2 (1,5,8,2) 16 5 N N N 21 10 30 (40) 229 174 N/A [123,124,15,25,34,35] 2 (1,5,9,2) 17 6 N N N 20 11 36 (47) 313 332 N/A [123,124,35,45] 2 (1,5,7,2) 15 4 N N N 22 9 20 (36) 37 12 N/A [123,124,25,35,45] 2 (1,5,8,2) 16 5 N N N 21 10 26 (42) 89 61 N/A [123,124,15,25,35,45] 2 (1,5,9,2) 17 6 N N N 20 11 35 (48) 157 168 N/A [123,124,34,35,45] 2 (1,5,8,2) 16 5 N N N 21 10 28 (41) 113 121 N/A [123,124,15,34,35,45] 2 (1,5,9,2) 17 6 N N N 20 11 34 (48) 253 296 N/A [123,124,15,25,34,35,45] 2 (1,5,10,2) 18 7 N N N 19 12 43 (55) 441 602 N/A [123,124,125] 2 (1,5,7,3) 16 3 Y Y Y 21 10 20 (46) 25 8 N/A [123,124,125,35] 2 (1,5,8,3) 17 4 N N N 20 11 32 (50) 149 132 N/A [123,124,125,35,45] 2 (1,5,9,3) 18 5 N N N 19 12 38 (58) 129 142 N/A [123,124,125,34,35,45] 2 (1,5,10,3) 19 6 N N N 18 13 46 (66) 333 400 N/A [125,134] 2 (1,5,6,2) 14 2 Y N N 23 8 20 (27) 113 55 N/A [125,134,35] 2 (1,5,7,2) 15 3 N N N 22 9 24 (33) 165 118 N/A [125,134,35,45] 2 (1,5,8,2) 16 4 N N N 21 10 31 (39) 265 244 N/A [125,134,23,45] 2 (1,5,8,2) 16 4 N N N 21 10 28 (40) 265 153 N/A [125,134,23,35,45] 2 (1,5,9,2) 17 5 N N N 20 11 38 (46) 437 476 N/A [123,125,134] 2 (1,5,7,3) 16 3 Y N N 21 10 25 (43) 89 33 N/A [123,125,134,35] 2 (1,5,8,3) 17 4 N N N 20 11 33 (49) 177 140 N/A [123,125,134,45] 2 (1,5,8,3) 17 4 N N N 20 11 29 (51) 157 60 N/A [123,125,134,35,45] 2 (1,5,9,3) 18 5 N N N 19 12 40 (57) 317 281 N/A [123,125,134,24,35] 2 (1,5,9,3) 18 5 N N N 19 12 41 (56) 369 389 N/A [125,134,23,24,35,45] 2 (1,5,10,2) 18 6 N N N 19 12 48 (53) 577 561 N/A [123,125,134,24,35,45] 2 (1,5,10,3) 19 6 N N N 18 13 51 (64) 573 699 N/A [123,124,134,5] 2 (1,5,6,3) 15 4 N N N 22 9 20 (36) 89 46 N/A [123,124,134,35] 2 (1,5,7,3) 16 4 N N N 21 10 26 (42) 129 62 N/A [123,124,134,15] 2 (1,5,7,3) 16 4 N N N 21 10 25 (43) 93 50 N/A [123,124,134,15,35] 2 (1,5,8,3) 17 5 N N N 20 11 33 (49) 273 190 N/A [123,124,134,25,35] 2 (1,5,8,3) 17 5 N N N 20 11 31 (50) 117 79 N/A [123,124,134,15,25,35] 2 (1,5,9,3) 18 6 N N N 19 12 40 (57) 285 295 N/A [123,124,125,134] 2 (1,5,8,4) 18 4 Y N N 19 12 35 (60) 145 70 N/A [123,124,125,134,35] 2 (1,5,9,4) 19 5 N N N 18 13 43 (68) 281 219 N/A [123,124,134,25,35,45] 2 (1,5,9,3) 18 6 N N N 19 12 37 (58) 289 166 N/A [123,124,134,15,25,35,45] 2 (1,5,10,3) 19 7 N N N 18 13 49 (65) 489 504 N/A [123,124,125,134,35,45] 2 (1,5,10,4) 20 6 N N N 17 14 54 (76) 441 461 N/A [124,125,134,135] 2 (1,5,8,4) 18 4 Y Y Y 19 12 30 (63) 49 15 N/A [124,125,134,135,23] 2 (1,5,9,4) 19 5 N N N 18 13 42 (69) 137 147 N/A [123,124,125,134,135] 2 (1,5,9,5) 20 5 Y Y Y 17 14 45 (81) 133 85 N/A [124,125,134,135,23,45] 2 (1,5,10,4) 20 6 N N N 17 14 54 (76) 417 521 N/A [123,124,125,134,135,45] 2 (1,5,10,5) 21 6 N N N 16 15 57 (89) 389 399 N/A [123,124,125,134,135,145] 2 (1,5,10,6) 22 6 Y Y N 15 16 60 (103) 337 189 N/A [123,124,134,234,5] 2 (1,5,6,4) 16 5 N N N 21 10 23 (44) 169 93 N/A [123,124,134,234,25] 2 (1,5,7,4) 17 5 N N N 20 11 29 (51) 201 105 N/A [125,134,234] 2 (1,5,8,3) 17 3 Y N N 20 11 34 (48) 277 194 N/A [123,125,134,234] 2 (1,5,8,4) 18 4 Y N N 19 12 36 (59) 217 94 N/A [123,124,134,15,234,25] 2 (1,5,8,4) 18 6 N N N 19 12 36 (59) 249 200 N/A [123,124,125,134,234] 2 (1,5,8,5) 19 5 Y N N 18 13 39 (70) 269 133 N/A [125,134,234,45] 2 (1,5,9,3) 18 4 N N N 19 12 43 (55) 461 463 N/A [124,125,134,234,45] 2 (1,5,9,4) 19 5 N N N 18 13 47 (66) 469 364 N/A [123,125,134,234,45] 2 (1,5,9,4) 19 5 N N N 18 13 45 (67) 369 203 N/A [123,124,134,15,234,25,45] 2 (1,5,9,4) 19 7 N N N 18 13 45 (67) 545 454 N/A [123,124,125,134,234,45] 2 (1,5,9,5) 20 6 N N N 17 14 50 (78) 505 372 N/A [123,125,145,234] 2 (1,5,9,4) 19 4 Y N N 18 13 46 (66) 381 293 N/A [123,124,125,145,234] 2 (1,5,9,5) 20 5 Y N N 17 14 50 (78) 401 232 N/A [123,125,134,145,234] 2 (1,5,9,5) 20 5 Y N N 17 14 48 (79) 281 165 N/A [123,124,125,134,145,234] 2 (1,5,9,6) 21 6 Y N N 16 15 53 (91) 381 238 N/A [125,134,234,35,45] 2 (1,5,10,3) 19 5 N N N 18 13 57 (62) 725 923 N/A [123,125,134,234,35,45] 2 (1,5,10,4) 20 6 N N N 17 14 61 (74) 717 779+ N/A [125,134,145,234,35] 2 (1,5,10,4) 20 5 N N N 17 14 62 (73) 705 976+ N/A [123,125,134,145,234,35] 2 (1,5,10,5) 21 6 N N N 16 15 66 (86) 593 696 N/A [125,135,145,234] 2 (1,5,10,4) 20 4 Y N N 17 14 62 (73) 569 575 N/A [123,125,135,145,234] 2 (1,5,10,5) 21 5 Y N N 16 15 67 (85) 677 726 N/A [123,125,134,135,234,45] 2 (1,5,10,5) 21 6 N N N 16 15 66 (86) 693 652+ N/A [123,125,134,135,145,234] 2 (1,5,10,6) 22 6 Y N N 15 16 71 (99) 601 548+ N/A [123,124,134,15,234,25,35,45] 2 (1,5,10,4) 20 8 N N N 17 14 57 (75) 649 812 N/A [123,124,134,145,234,25,35] 2 (1,5,10,5) 21 7 N N N 16 15 66 (86) 853 1046+ N/A [123,124,125,134,145,234,35] 2 (1,5,10,6) 22 7 N N N 15 16 71 (99) 749 867+ N/A [123,124,125,134,135,145,234] 2 (1,5,10,7) 23 7 Y N N 14 17 76 (113) 829 708+ N/A [124,125,134,135,234,235] 2 (1,5,9,6) 21 6 Y Y Y 16 15 50 (93) 145 61 N/A [123,124,125,134,135,234,235] 2 (1,5,9,7) 22 7 Y Y Y 15 16 56 (105) 253+ 180 N/A [124,125,134,135,234,235,45] 2 (1,5,10,6) 22 7 N N N 15 16 70 (100) 457 547+ N/A [123,124,125,134,135,234,235,45] 2 (1,5,10,7) 23 8 N N N 14 17 76 (113) 757 1064 N/A [123,124,134,145,234,235] 2 (1,5,10,6) 22 6 Y N N 15 16 73 (97,1) 897 987 N/A [125,134,145,234,235] 2 (1,5,10,5) 21 5 Y N N 16 15 68 (84) 765 927 N/A [123,125,134,145,234,235] 2 (1,5,10,6) 22 6 Y N N 15 16 72 (98) 649 626 N/A [123,124,125,134,145,234,235] 2 (1,5,10,7) 23 7 Y N N 14 17 77 (112) 725 730 N/A [124,125,134,135,145,234,235] 2 (1,5,10,7) 23 7 Y N N 14 17 76 (113) 501 481 N/A [123,124,125,134,135,145,234,235] 2 (1,5,10,8) 24 8 Y N N 13 18 82 (127) 857 810 N/A [123,124,134,135,145,234,235,245] 2 (1,5,10,8) 24 8 Y Y Y 13 18 81 (128) 449 485+ N/A [123,124,125,134,135,145,234,235,245] 2 (1,5,10,9) 25 9 Y Y N 12 19 87 (143) 813 774 N/A [123,124,125,134,135,145,234,235,245,345] 2 (1,5,10,10) 26 10 Y Y N 11 20 93 (159) 1425 1247 N/A [1234,5] 3 (1,5,6,4,1) 17 2 N N N 20 11 24 (55) 73 24 N/A [1234,15] 3 (1,5,7,4,1) 18 2 N N N 19 12 30 (63) 81 28 N/A [1234,15,25] 3 (1,5,8,4,1) 19 3 N N N 18 13 38 (71) 217 94 N/A [1234,125] 3 (1,5,8,5,1) 20 2 N N N 17 14 40 (84) 101 36 N/A [1234,15,25,35] 3 (1,5,9,4,1) 20 4 N N N 17 14 49 (79) 401 220 N/A [1234,135,25] 3 (1,5,9,5,1) 21 3 N N N 16 15 52 (92) 281 178 N/A [1234,135,235] 3 (1,5,9,6,1) 22 3 N N N 15 16 55 (106) 249 100 N/A [1234,125,135,235] 3 (1,5,9,7,1) 23 4 N N N 14 17 59 (120) 417 171 N/A [1234,15,25,35,45] 3 (1,5,10,4,1) 21 5 N N N 16 15 65 (87) 681 605+ N/A [1234,15,235,45] 3 (1,5,10,5,1) 22 4 N N N 15 16 70 (100) 517 534 N/A [1234,125,235,45] 3 (1,5,10,6,1) 23 4 N N N 14 17 75 (114) 541 539 N/A [1234,145,235] 3 (1,5,10,6,1) 23 3 N N N 14 17 76 (113) 449 473 N/A [1234,125,145,235] 3 (1,5,10,7,1) 24 4 N N N 13 18 81 (128) 613 469 N/A [1234,125,135,145] 3 (1,5,10,7,1) 24 4 N N N 13 18 80 (129) 505 284 N/A [1234,125,135,235,45] 3 (1,5,10,7,1) 24 5 N N N 13 18 82 (128) 885 784 N/A [1234,125,135,145,235] 3 (1,5,10,8,1) 25 5 N N N 12 19 88 (143) 865 654 N/A [1234,125,145,235,345] 3 (1,5,10,8,1) 25 5 N N N 12 19 86 (144) 689 445 N/A [1234,125,135,145,235,345] 3 (1,5,10,9,1) 26 6 N N N 11 20 95 (159) 1005 710 N/A [1234,125,135,145,235,245,345] 3 (1,5,10,10,1) 27 7 N N N 10 21 104 (175) 1457 1077 N/A [1234,1235] 3 (1,5,9,7,2) 24 2 Y Y Y 13 18 60 (138) 133 42 N/A [1234,1235,45] 3 (1,5,10,7,2) 25 3 N N N 12 19 84 (146) 401 432 N/A [1234,1235,245] 3 (1,5,10,8,2) 26 3 N N N 11 20 90 (162) 433 262 N/A [1234,1235,145,245] 3 (1,5,10,9,2) 27 4 N N N 10 21 98 (178) 753 530 N/A [1234,1235,145,245,345] 3 (1,5,10,10,2) 28 5 N N N 9 22 109 (194) 1153 818 N/A [1234,1235,1245] 3 (1,5,10,9,3) 28 3 Y Y Y 9 22 100 (199) 409 176 N/A [1234,1235,1245,345] 3 (1,5,10,10,3) 29 4 N N N 8 23 112 (215) 825 556 N/A [1234,1235,1245,1345] 3 (1,5,10,10,4) 30 4 Y Y Y 7 24 115 (237) 725 298 N/A [1234,1235,1245,1345,2345] 3 (1,5,10,10,5) 31 5 Y Y Y 6 25 119 (259) 1137 485 N/A [12345] 4 (1,5,10,10,5,1) 32 1 Y Y Y 5 26 120 (285) 369 120 N/A