A table of some Kostka numbers The rows and columns are indexed by partitions in reverse lex order. Before each matrix I've listed the partitions in that order; the columns go across, and the rows go down (i.e., the only '2' in the n = 3 Kostka matrix is in row {2,1} and column {1,1,1}). The entry in row lambda and column mu is the Kostka number for shape lambda and content mu. Calculated using Mathematica and Tableaux.m from http://www.math.umn.edu/~drake/ n = 2 ------------------------------------------- {{2}, {1, 1}} 1 1 0 1 n = 3 ------------------------------------------- {{3}, {2, 1}, {1, 1, 1}} 1 1 1 0 1 2 0 0 1 n = 4 ------------------------------------------- {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}} 1 1 1 1 1 0 1 1 2 3 0 0 1 1 2 0 0 0 1 3 0 0 0 0 1 n = 5 ------------------------------------------- {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}} 1 1 1 1 1 1 1 0 1 1 2 2 3 4 0 0 1 1 2 3 5 0 0 0 1 1 3 6 0 0 0 0 1 2 5 0 0 0 0 0 1 4 0 0 0 0 0 0 1 n = 6 ------------------------------------------- {{6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}} 1 1 1 1 1 1 1 1 1 1 1 0 1 1 2 1 2 3 2 3 4 5 0 0 1 1 1 2 3 3 4 6 9 0 0 0 1 0 1 3 1 3 6 10 0 0 0 0 1 1 1 1 2 3 5 0 0 0 0 0 1 2 2 4 8 16 0 0 0 0 0 0 1 0 1 4 10 0 0 0 0 0 0 0 1 1 2 5 0 0 0 0 0 0 0 0 1 3 9 0 0 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 0 0 0 0 1 n = 7 ------------------------------------------- {{7}, {6, 1}, {5, 2}, {5, 1, 1}, {4, 3}, {4, 2, 1}, {4, 1, 1, 1}, {3, 3, 1}, {3, 2, 2}, {3, 2, 1, 1}, {3, 1, 1, 1, 1}, {2, 2, 2, 1}, {2, 2, 1, 1, 1}, {2, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1}} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 2 1 2 3 2 2 3 4 3 4 5 6 0 0 1 1 1 2 3 2 3 4 6 5 7 10 14 0 0 0 1 0 1 3 1 1 3 6 3 6 10 15 0 0 0 0 1 1 1 2 2 3 4 4 6 9 14 0 0 0 0 0 1 2 1 2 4 8 6 11 20 35 0 0 0 0 0 0 1 0 0 1 4 1 4 10 20 0 0 0 0 0 0 0 1 1 2 3 3 6 11 21 0 0 0 0 0 0 0 0 1 1 2 3 5 10 21 0 0 0 0 0 0 0 0 0 1 3 2 6 15 35 0 0 0 0 0 0 0 0 0 0 1 0 1 5 15 0 0 0 0 0 0 0 0 0 0 0 1 2 5 14 0 0 0 0 0 0 0 0 0 0 0 0 1 4 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n = 7 ------------------------------------------- {{8}, {7, 1}, {6, 2}, {6, 1, 1}, {5, 3}, {5, 2, 1}, {5, 1, 1, 1}, {4, 4}, {4, 3, 1}, {4, 2, 2}, {4, 2, 1, 1}, {4, 1, 1, 1, 1}, {3, 3, 2}, {3, 3, 1, 1}, {3, 2, 2, 1}, {3, 2, 1, 1, 1}, {3, 1, 1, 1, 1, 1}, {2, 2, 2, 2}, {2, 2, 2, 1, 1}, {2, 2, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 2 1 2 3 1 2 2 3 4 2 3 3 4 5 3 4 5 6 7 0 0 1 1 1 2 3 1 2 3 4 6 3 4 5 7 10 6 8 11 15 20 0 0 0 1 0 1 3 0 1 1 3 6 1 3 3 6 10 3 6 10 15 21 0 0 0 0 1 1 1 1 2 2 3 4 3 4 5 7 10 6 9 13 19 28 0 0 0 0 0 1 2 0 1 2 4 8 2 4 6 11 20 8 14 24 40 64 0 0 0 0 0 0 1 0 0 0 1 4 0 1 1 4 10 1 4 10 20 35 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 3 4 3 4 6 9 14 0 0 0 0 0 0 0 0 1 1 2 3 2 4 5 9 15 7 13 23 40 70 0 0 0 0 0 0 0 0 0 1 1 2 1 1 3 5 10 6 9 16 30 56 0 0 0 0 0 0 0 0 0 0 1 3 0 1 2 6 15 3 9 21 45 90 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 5 0 1 5 15 35 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 3 6 11 21 42 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 6 2 5 12 26 56 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 3 6 13 30 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 2 8 24 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 6 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n = 9 ------------------------------------------- {{9}, {8, 1}, {7, 2}, {7, 1, 1}, {6, 3}, {6, 2, 1}, {6, 1, 1, 1}, {5, 4}, {5, 3, 1}, {5, 2, 2}, {5, 2, 1, 1}, {5, 1, 1, 1, 1}, {4, 4, 1}, {4, 3, 2}, {4, 3, 1, 1}, {4, 2, 2, 1}, {4, 2, 1, 1, 1}, {4, 1, 1, 1, 1, 1}, {3, 3, 3}, {3, 3, 2, 1}, {3, 3, 1, 1, 1}, {3, 2, 2, 2}, {3, 2, 2, 1, 1}, {3, 2, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 1}, {2, 2, 2, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1}} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 2 1 2 3 1 2 2 3 4 2 2 3 3 4 5 2 3 4 3 4 5 6 4 5 6 7 8 0 0 1 1 1 2 3 1 2 3 4 6 2 3 4 5 7 10 3 5 7 6 8 11 15 9 12 16 21 27 0 0 0 1 0 1 3 0 1 1 3 6 1 1 3 3 6 10 1 3 6 3 6 10 15 6 10 15 21 28 0 0 0 0 1 1 1 1 2 2 3 4 2 3 4 5 7 10 4 6 8 7 10 14 20 12 17 24 34 48 0 0 0 0 0 1 2 0 1 2 4 8 1 2 4 6 11 20 2 6 11 8 14 24 40 17 28 45 70 105 0 0 0 0 0 0 1 0 0 0 1 4 0 0 1 1 4 10 0 1 4 1 4 10 20 4 10 20 35 56 0 0 0 0 0 0 0 1 1 1 1 1 2 2 3 3 4 5 2 4 6 5 7 10 14 9 13 19 28 42 0 0 0 0 0 0 0 0 1 1 2 3 1 2 4 5 9 15 3 7 12 9 16 27 45 21 36 60 99 162 0 0 0 0 0 0 0 0 0 1 1 2 0 1 1 3 5 10 1 3 5 6 9 16 30 14 23 40 70 120 0 0 0 0 0 0 0 0 0 0 1 3 0 0 1 2 6 15 0 2 6 3 9 21 45 12 27 55 105 189 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 5 0 0 1 0 1 5 15 1 5 15 35 70 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 4 1 3 6 4 7 12 19 10 17 29 49 84 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 2 4 6 6 10 17 30 16 28 50 91 168 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 6 0 2 6 3 8 18 36 12 27 56 111 216 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 0 1 2 3 6 13 30 12 24 50 105 216 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 1 0 2 8 24 3 12 34 84 189 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 6 0 1 6 21 56 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 3 5 3 6 11 21 42 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 4 8 16 8 17 36 77 168 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 4 10 2 7 20 50 120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 4 7 15 35 84 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 3 9 24 63 162 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 0 2 10 35 105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 7 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 14 42 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 14 48 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 n = 10 ------------------------------------------- {{10}, {9, 1}, {8, 2}, {8, 1, 1}, {7, 3}, {7, 2, 1}, {7, 1, 1, 1}, {6, 4}, {6, 3, 1}, {6, 2, 2}, {6, 2, 1, 1}, {6, 1, 1, 1, 1}, {5, 5}, {5, 4, 1}, {5, 3, 2}, {5, 3, 1, 1}, {5, 2, 2, 1}, {5, 2, 1, 1, 1}, {5, 1, 1, 1, 1, 1}, {4, 4, 2}, {4, 4, 1, 1}, {4, 3, 3}, {4, 3, 2, 1}, {4, 3, 1, 1, 1}, {4, 2, 2, 2}, {4, 2, 2, 1, 1}, {4, 2, 1, 1, 1, 1}, {4, 1, 1, 1, 1, 1, 1}, {3, 3, 3, 1}, {3, 3, 2, 2}, {3, 3, 2, 1, 1}, {3, 3, 1, 1, 1, 1}, {3, 2, 2, 2, 1}, {3, 2, 2, 1, 1, 1}, {3, 2, 1, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2}, {2, 2, 2, 2, 1, 1}, {2, 2, 2, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}} (This one's 106 columns wide.) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 2 1 2 3 1 2 2 3 4 1 2 2 3 3 4 5 2 3 2 3 4 3 4 5 6 3 3 4 5 4 5 6 7 4 5 6 7 8 9 0 0 1 1 1 2 3 1 2 3 4 6 1 2 3 4 5 7 10 3 4 3 5 7 6 8 11 15 5 6 8 11 9 12 16 21 10 13 17 22 28 35 0 0 0 1 0 1 3 0 1 1 3 6 0 1 1 3 3 6 10 1 3 1 3 6 3 6 10 15 3 3 6 10 6 10 15 21 6 10 15 21 28 36 0 0 0 0 1 1 1 1 2 2 3 4 1 2 3 4 5 7 10 3 4 4 6 8 7 10 14 20 7 8 11 15 13 18 25 35 15 21 29 40 55 75 0 0 0 0 0 1 2 0 1 2 4 8 0 1 2 4 6 11 20 2 4 2 6 11 8 14 24 40 6 8 14 24 17 28 45 70 20 32 50 76 112 160 0 0 0 0 0 0 1 0 0 0 1 4 0 0 0 1 1 4 10 0 1 0 1 4 1 4 10 20 1 1 4 10 4 10 20 35 4 10 20 35 56 84 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 3 3 4 5 3 4 3 5 7 6 8 11 15 6 7 10 14 12 17 24 34 15 21 30 43 62 90 0 0 0 0 0 0 0 0 1 1 2 3 0 1 2 4 5 9 15 2 4 3 7 12 9 16 27 45 9 11 19 31 24 40 65 105 30 50 81 129 203 315 0 0 0 0 0 0 0 0 0 1 1 2 0 0 1 1 3 5 10 1 1 1 3 5 6 9 16 30 3 6 9 16 14 23 40 70 20 31 51 85 140 225 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 1 2 6 15 0 1 0 2 6 3 9 21 45 2 3 9 21 12 27 55 105 15 33 65 120 210 350 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 5 0 0 0 0 1 0 1 5 15 0 0 1 5 1 5 15 35 1 5 15 35 70 126 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 1 2 3 2 3 4 5 2 3 4 6 5 7 10 14 6 9 13 19 28 42 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 4 2 4 2 5 9 6 10 16 24 6 8 14 24 18 30 49 78 24 40 66 108 176 288 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 1 1 2 4 6 6 10 17 30 6 9 14 23 21 35 60 105 30 51 87 150 260 450 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 6 0 1 0 2 6 3 8 18 36 3 4 11 24 15 33 66 126 21 45 90 171 315 567 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 0 0 0 1 2 3 6 13 30 1 3 6 13 12 24 50 105 20 38 74 145 280 525 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 0 1 0 2 8 24 0 0 2 8 3 12 34 84 4 16 44 104 224 448 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 6 0 0 0 1 0 1 6 21 0 1 6 21 56 126 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 4 6 9 3 4 7 12 10 17 29 49 16 26 45 79 140 252 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 1 3 6 10 1 2 5 12 7 15 30 55 10 22 44 85 160 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 3 5 3 3 5 7 7 12 20 35 10 19 34 61 112 210 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 4 8 16 2 4 8 16 14 28 56 112 24 48 96 192 384 768 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 4 10 0 0 2 8 3 11 30 70 5 17 46 110 245 525 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 0 1 1 2 4 7 15 35 10 16 31 65 140 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 0 0 1 3 3 9 24 63 6 18 45 108 252 567 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 0 0 0 1 0 2 10 35 0 3 15 50 140 350 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 7 0 0 1 7 28 84 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 11 21 5 11 23 47 98 210 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 10 21 6 12 24 51 112 252 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 2 6 15 35 5 13 33 80 190 450 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 5 15 0 2 9 30 85 225 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 14 4 8 18 44 112 288 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 14 0 3 12 38 112 315 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 0 0 2 12 48 160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 14 42 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 28 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 20 75 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 calculated by Dan Drake (http://www.math.umn.edu/~drake)