All the Possible Score Vectors in a Round-Robin Chess Tournament, and the Nu\ mber of Times each Shows Up for tournaments from Three to , 9, players. By Shalosh B. Ekhad In his beautiful article, "Chess Tournament and the like treated by the cal\ culus of Symmetric Functions" Quarterly J. of Pure and Applied Mathematics. vol. 49 (1923), 353-384 The Combinatorial Giant, (and amazing human calculator), Major Percy Alexand\ er MacMahon considered round-robin Chess tournaments for n players for n from 2 to 6, an\ d for each found all the possible score vectors and for each their number of occurrences. In this article, we reproduce his tables, and go all the way to tournaments \ with, 9, players. Recall that in a Chess match with a winner, the winner gets one point, and t\ he loser gets 0 points, and in case of a draw, each get half a point. For the sake of conbenience let's double ever\ ythinng, so the score vectors below should be divided by 2. ----------------------------------------------------------------------------\ ----- 3, Players. Symmetric Function:, 3 / 3 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [4, 2, 0], 1, 6, 6 [4, 1, 1], 1, 3, 3 [3, 3, 0], 1, 3, 3 [3, 2, 1], 2, 6, 12 [2, 2, 2], 3, 1, 3 Sum of numbers in last column=, 27 The most probable vector is, [3, 2, 1], which arises in, 12, out of, 3, to the power, 3 that equals, 27, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 3, players The sole champion score-vector is, [3, 2, 1] the corresponding record is, 12 ----------------------------------------------------------------------------\ ----- 4, Players. Symmetric Function:, 4 / 4 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [6, 4, 2, 0], 1, 24, 24 [6, 4, 1, 1], 1, 12, 12 [6, 3, 3, 0], 1, 12, 12 [6, 3, 2, 1], 2, 24, 48 [6, 2, 2, 2], 3, 4, 12 [5, 5, 2, 0], 1, 12, 12 [5, 5, 1, 1], 1, 6, 6 [5, 4, 3, 0], 2, 24, 48 [5, 4, 2, 1], 4, 24, 96 [5, 3, 3, 1], 5, 12, 60 [5, 3, 2, 2], 7, 12, 84 [4, 4, 4, 0], 3, 4, 12 [4, 4, 3, 1], 7, 12, 84 [4, 4, 2, 2], 10, 6, 60 [4, 3, 3, 2], 12, 12, 144 [3, 3, 3, 3], 15, 1, 15 Sum of numbers in last column=, 729 The most probable vector is, [4, 3, 3, 2], which arises in, 144, out of, 3, to the power, 6 that equals, 729, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 4, players The sole champion score-vector is, [4, 3, 3, 2] the corresponding record is, 144 ----------------------------------------------------------------------------\ ----- 5, Players. Symmetric Function:, 5 / 5 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [8, 6, 4, 2, 0], 1, 120, 120 [8, 6, 4, 1, 1], 1, 60, 60 [8, 6, 3, 3, 0], 1, 60, 60 [8, 6, 3, 2, 1], 2, 120, 240 [8, 6, 2, 2, 2], 3, 20, 60 [8, 5, 5, 2, 0], 1, 60, 60 [8, 5, 5, 1, 1], 1, 30, 30 [8, 5, 4, 3, 0], 2, 120, 240 [8, 5, 4, 2, 1], 4, 120, 480 [8, 5, 3, 3, 1], 5, 60, 300 [8, 5, 3, 2, 2], 7, 60, 420 [8, 4, 4, 4, 0], 3, 20, 60 [8, 4, 4, 3, 1], 7, 60, 420 [8, 4, 4, 2, 2], 10, 30, 300 [8, 4, 3, 3, 2], 12, 60, 720 [8, 3, 3, 3, 3], 15, 5, 75 [7, 7, 4, 2, 0], 1, 60, 60 [7, 7, 4, 1, 1], 1, 30, 30 [7, 7, 3, 3, 0], 1, 30, 30 [7, 7, 3, 2, 1], 2, 60, 120 [7, 7, 2, 2, 2], 3, 10, 30 [7, 6, 5, 2, 0], 2, 120, 240 [7, 6, 5, 1, 1], 2, 60, 120 [7, 6, 4, 3, 0], 4, 120, 480 [7, 6, 4, 2, 1], 8, 120, 960 [7, 6, 3, 3, 1], 10, 60, 600 [7, 6, 3, 2, 2], 14, 60, 840 [7, 5, 5, 3, 0], 5, 60, 300 [7, 5, 5, 2, 1], 10, 60, 600 [7, 5, 4, 4, 0], 7, 60, 420 [7, 5, 4, 3, 1], 18, 120, 2160 [7, 5, 4, 2, 2], 25, 60, 1500 [7, 5, 3, 3, 2], 31, 60, 1860 [7, 4, 4, 4, 1], 24, 20, 480 [7, 4, 4, 3, 2], 41, 60, 2460 [7, 4, 3, 3, 3], 51, 20, 1020 [6, 6, 6, 2, 0], 3, 20, 60 [6, 6, 6, 1, 1], 3, 10, 30 [6, 6, 5, 3, 0], 7, 60, 420 [6, 6, 5, 2, 1], 14, 60, 840 [6, 6, 4, 4, 0], 10, 30, 300 [6, 6, 4, 3, 1], 25, 60, 1500 [6, 6, 4, 2, 2], 35, 30, 1050 [6, 6, 3, 3, 2], 43, 30, 1290 [6, 5, 5, 4, 0], 12, 60, 720 [6, 5, 5, 3, 1], 31, 60, 1860 [6, 5, 5, 2, 2], 43, 30, 1290 [6, 5, 4, 4, 1], 41, 60, 2460 [6, 5, 4, 3, 2], 70, 120, 8400 [6, 5, 3, 3, 3], 87, 20, 1740 [6, 4, 4, 4, 2], 90, 20, 1800 [6, 4, 4, 3, 3], 111, 30, 3330 [5, 5, 5, 5, 0], 15, 5, 75 [5, 5, 5, 4, 1], 51, 20, 1020 [5, 5, 5, 3, 2], 87, 20, 1740 [5, 5, 4, 4, 2], 111, 30, 3330 [5, 5, 4, 3, 3], 138, 30, 4140 [5, 4, 4, 4, 3], 174, 20, 3480 [4, 4, 4, 4, 4], 219, 1, 219 Sum of numbers in last column=, 59049 The most probable vector is, [6, 5, 4, 3, 2], which arises in, 8400, out of, 3, to the power, 10 that equals, 59049, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 5, players The sole champion score-vector is, [6, 5, 4, 3, 2] the corresponding record is, 8400 ----------------------------------------------------------------------------\ ----- 6, Players. Symmetric Function:, 6 / 6 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [10, 8, 6, 4, 2, 0], 1, 720, 720 [10, 8, 6, 4, 1, 1], 1, 360, 360 [10, 8, 6, 3, 3, 0], 1, 360, 360 [10, 8, 6, 3, 2, 1], 2, 720, 1440 [10, 8, 6, 2, 2, 2], 3, 120, 360 [10, 8, 5, 5, 2, 0], 1, 360, 360 [10, 8, 5, 5, 1, 1], 1, 180, 180 [10, 8, 5, 4, 3, 0], 2, 720, 1440 [10, 8, 5, 4, 2, 1], 4, 720, 2880 [10, 8, 5, 3, 3, 1], 5, 360, 1800 [10, 8, 5, 3, 2, 2], 7, 360, 2520 [10, 8, 4, 4, 4, 0], 3, 120, 360 [10, 8, 4, 4, 3, 1], 7, 360, 2520 [10, 8, 4, 4, 2, 2], 10, 180, 1800 [10, 8, 4, 3, 3, 2], 12, 360, 4320 [10, 8, 3, 3, 3, 3], 15, 30, 450 [10, 7, 7, 4, 2, 0], 1, 360, 360 [10, 7, 7, 4, 1, 1], 1, 180, 180 [10, 7, 7, 3, 3, 0], 1, 180, 180 [10, 7, 7, 3, 2, 1], 2, 360, 720 [10, 7, 7, 2, 2, 2], 3, 60, 180 [10, 7, 6, 5, 2, 0], 2, 720, 1440 [10, 7, 6, 5, 1, 1], 2, 360, 720 [10, 7, 6, 4, 3, 0], 4, 720, 2880 [10, 7, 6, 4, 2, 1], 8, 720, 5760 [10, 7, 6, 3, 3, 1], 10, 360, 3600 [10, 7, 6, 3, 2, 2], 14, 360, 5040 [10, 7, 5, 5, 3, 0], 5, 360, 1800 [10, 7, 5, 5, 2, 1], 10, 360, 3600 [10, 7, 5, 4, 4, 0], 7, 360, 2520 [10, 7, 5, 4, 3, 1], 18, 720, 12960 [10, 7, 5, 4, 2, 2], 25, 360, 9000 [10, 7, 5, 3, 3, 2], 31, 360, 11160 [10, 7, 4, 4, 4, 1], 24, 120, 2880 [10, 7, 4, 4, 3, 2], 41, 360, 14760 [10, 7, 4, 3, 3, 3], 51, 120, 6120 [10, 6, 6, 6, 2, 0], 3, 120, 360 [10, 6, 6, 6, 1, 1], 3, 60, 180 [10, 6, 6, 5, 3, 0], 7, 360, 2520 [10, 6, 6, 5, 2, 1], 14, 360, 5040 [10, 6, 6, 4, 4, 0], 10, 180, 1800 [10, 6, 6, 4, 3, 1], 25, 360, 9000 [10, 6, 6, 4, 2, 2], 35, 180, 6300 [10, 6, 6, 3, 3, 2], 43, 180, 7740 [10, 6, 5, 5, 4, 0], 12, 360, 4320 [10, 6, 5, 5, 3, 1], 31, 360, 11160 [10, 6, 5, 5, 2, 2], 43, 180, 7740 [10, 6, 5, 4, 4, 1], 41, 360, 14760 [10, 6, 5, 4, 3, 2], 70, 720, 50400 [10, 6, 5, 3, 3, 3], 87, 120, 10440 [10, 6, 4, 4, 4, 2], 90, 120, 10800 [10, 6, 4, 4, 3, 3], 111, 180, 19980 [10, 5, 5, 5, 5, 0], 15, 30, 450 [10, 5, 5, 5, 4, 1], 51, 120, 6120 [10, 5, 5, 5, 3, 2], 87, 120, 10440 [10, 5, 5, 4, 4, 2], 111, 180, 19980 [10, 5, 5, 4, 3, 3], 138, 180, 24840 [10, 5, 4, 4, 4, 3], 174, 120, 20880 [10, 4, 4, 4, 4, 4], 219, 6, 1314 [9, 9, 6, 4, 2, 0], 1, 360, 360 [9, 9, 6, 4, 1, 1], 1, 180, 180 [9, 9, 6, 3, 3, 0], 1, 180, 180 [9, 9, 6, 3, 2, 1], 2, 360, 720 [9, 9, 6, 2, 2, 2], 3, 60, 180 [9, 9, 5, 5, 2, 0], 1, 180, 180 [9, 9, 5, 5, 1, 1], 1, 90, 90 [9, 9, 5, 4, 3, 0], 2, 360, 720 [9, 9, 5, 4, 2, 1], 4, 360, 1440 [9, 9, 5, 3, 3, 1], 5, 180, 900 [9, 9, 5, 3, 2, 2], 7, 180, 1260 [9, 9, 4, 4, 4, 0], 3, 60, 180 [9, 9, 4, 4, 3, 1], 7, 180, 1260 [9, 9, 4, 4, 2, 2], 10, 90, 900 [9, 9, 4, 3, 3, 2], 12, 180, 2160 [9, 9, 3, 3, 3, 3], 15, 15, 225 [9, 8, 7, 4, 2, 0], 2, 720, 1440 [9, 8, 7, 4, 1, 1], 2, 360, 720 [9, 8, 7, 3, 3, 0], 2, 360, 720 [9, 8, 7, 3, 2, 1], 4, 720, 2880 [9, 8, 7, 2, 2, 2], 6, 120, 720 [9, 8, 6, 5, 2, 0], 4, 720, 2880 [9, 8, 6, 5, 1, 1], 4, 360, 1440 [9, 8, 6, 4, 3, 0], 8, 720, 5760 [9, 8, 6, 4, 2, 1], 16, 720, 11520 [9, 8, 6, 3, 3, 1], 20, 360, 7200 [9, 8, 6, 3, 2, 2], 28, 360, 10080 [9, 8, 5, 5, 3, 0], 10, 360, 3600 [9, 8, 5, 5, 2, 1], 20, 360, 7200 [9, 8, 5, 4, 4, 0], 14, 360, 5040 [9, 8, 5, 4, 3, 1], 36, 720, 25920 [9, 8, 5, 4, 2, 2], 50, 360, 18000 [9, 8, 5, 3, 3, 2], 62, 360, 22320 [9, 8, 4, 4, 4, 1], 48, 120, 5760 [9, 8, 4, 4, 3, 2], 82, 360, 29520 [9, 8, 4, 3, 3, 3], 102, 120, 12240 [9, 7, 7, 5, 2, 0], 5, 360, 1800 [9, 7, 7, 5, 1, 1], 5, 180, 900 [9, 7, 7, 4, 3, 0], 10, 360, 3600 [9, 7, 7, 4, 2, 1], 20, 360, 7200 [9, 7, 7, 3, 3, 1], 25, 180, 4500 [9, 7, 7, 3, 2, 2], 35, 180, 6300 [9, 7, 6, 6, 2, 0], 7, 360, 2520 [9, 7, 6, 6, 1, 1], 7, 180, 1260 [9, 7, 6, 5, 3, 0], 18, 720, 12960 [9, 7, 6, 5, 2, 1], 36, 720, 25920 [9, 7, 6, 4, 4, 0], 25, 360, 9000 [9, 7, 6, 4, 3, 1], 65, 720, 46800 [9, 7, 6, 4, 2, 2], 90, 360, 32400 [9, 7, 6, 3, 3, 2], 112, 360, 40320 [9, 7, 5, 5, 4, 0], 31, 360, 11160 [9, 7, 5, 5, 3, 1], 82, 360, 29520 [9, 7, 5, 5, 2, 2], 113, 180, 20340 [9, 7, 5, 4, 4, 1], 108, 360, 38880 [9, 7, 5, 4, 3, 2], 185, 720, 133200 [9, 7, 5, 3, 3, 3], 231, 120, 27720 [9, 7, 4, 4, 4, 2], 237, 120, 28440 [9, 7, 4, 4, 3, 3], 295, 180, 53100 [9, 6, 6, 6, 3, 0], 24, 120, 2880 [9, 6, 6, 6, 2, 1], 48, 120, 5760 [9, 6, 6, 5, 4, 0], 41, 360, 14760 [9, 6, 6, 5, 3, 1], 108, 360, 38880 [9, 6, 6, 5, 2, 2], 149, 180, 26820 [9, 6, 6, 4, 4, 1], 142, 180, 25560 [9, 6, 6, 4, 3, 2], 243, 360, 87480 [9, 6, 6, 3, 3, 3], 303, 60, 18180 [9, 6, 5, 5, 5, 0], 51, 120, 6120 [9, 6, 5, 5, 4, 1], 176, 360, 63360 [9, 6, 5, 5, 3, 2], 301, 360, 108360 [9, 6, 5, 4, 4, 2], 382, 360, 137520 [9, 6, 5, 4, 3, 3], 476, 360, 171360 [9, 6, 4, 4, 4, 3], 597, 120, 71640 [9, 5, 5, 5, 5, 1], 219, 30, 6570 [9, 5, 5, 5, 4, 2], 471, 120, 56520 [9, 5, 5, 5, 3, 3], 588, 60, 35280 [9, 5, 5, 4, 4, 3], 735, 180, 132300 [9, 5, 4, 4, 4, 4], 915, 30, 27450 [8, 8, 8, 4, 2, 0], 3, 120, 360 [8, 8, 8, 4, 1, 1], 3, 60, 180 [8, 8, 8, 3, 3, 0], 3, 60, 180 [8, 8, 8, 3, 2, 1], 6, 120, 720 [8, 8, 8, 2, 2, 2], 9, 20, 180 [8, 8, 7, 5, 2, 0], 7, 360, 2520 [8, 8, 7, 5, 1, 1], 7, 180, 1260 [8, 8, 7, 4, 3, 0], 14, 360, 5040 [8, 8, 7, 4, 2, 1], 28, 360, 10080 [8, 8, 7, 3, 3, 1], 35, 180, 6300 [8, 8, 7, 3, 2, 2], 49, 180, 8820 [8, 8, 6, 6, 2, 0], 10, 180, 1800 [8, 8, 6, 6, 1, 1], 10, 90, 900 [8, 8, 6, 5, 3, 0], 25, 360, 9000 [8, 8, 6, 5, 2, 1], 50, 360, 18000 [8, 8, 6, 4, 4, 0], 35, 180, 6300 [8, 8, 6, 4, 3, 1], 90, 360, 32400 [8, 8, 6, 4, 2, 2], 125, 180, 22500 [8, 8, 6, 3, 3, 2], 155, 180, 27900 [8, 8, 5, 5, 4, 0], 43, 180, 7740 [8, 8, 5, 5, 3, 1], 113, 180, 20340 [8, 8, 5, 5, 2, 2], 156, 90, 14040 [8, 8, 5, 4, 4, 1], 149, 180, 26820 [8, 8, 5, 4, 3, 2], 255, 360, 91800 [8, 8, 5, 3, 3, 3], 318, 60, 19080 [8, 8, 4, 4, 4, 2], 327, 60, 19620 [8, 8, 4, 4, 3, 3], 406, 90, 36540 [8, 7, 7, 6, 2, 0], 12, 360, 4320 [8, 7, 7, 6, 1, 1], 12, 180, 2160 [8, 7, 7, 5, 3, 0], 31, 360, 11160 [8, 7, 7, 5, 2, 1], 62, 360, 22320 [8, 7, 7, 4, 4, 0], 43, 180, 7740 [8, 7, 7, 4, 3, 1], 112, 360, 40320 [8, 7, 7, 4, 2, 2], 155, 180, 27900 [8, 7, 7, 3, 3, 2], 193, 180, 34740 [8, 7, 6, 6, 3, 0], 41, 360, 14760 [8, 7, 6, 6, 2, 1], 82, 360, 29520 [8, 7, 6, 5, 4, 0], 70, 720, 50400 [8, 7, 6, 5, 3, 1], 185, 720, 133200 [8, 7, 6, 5, 2, 2], 255, 360, 91800 [8, 7, 6, 4, 4, 1], 243, 360, 87480 [8, 7, 6, 4, 3, 2], 416, 720, 299520 [8, 7, 6, 3, 3, 3], 519, 120, 62280 [8, 7, 5, 5, 5, 0], 87, 120, 10440 [8, 7, 5, 5, 4, 1], 301, 360, 108360 [8, 7, 5, 5, 3, 2], 515, 360, 185400 [8, 7, 5, 4, 4, 2], 653, 360, 235080 [8, 7, 5, 4, 3, 3], 814, 360, 293040 [8, 7, 4, 4, 4, 3], 1020, 120, 122400 [8, 6, 6, 6, 4, 0], 90, 120, 10800 [8, 6, 6, 6, 3, 1], 237, 120, 28440 [8, 6, 6, 6, 2, 2], 327, 60, 19620 [8, 6, 6, 5, 5, 0], 111, 180, 19980 [8, 6, 6, 5, 4, 1], 382, 360, 137520 [8, 6, 6, 5, 3, 2], 653, 360, 235080 [8, 6, 6, 4, 4, 2], 826, 180, 148680 [8, 6, 6, 4, 3, 3], 1028, 180, 185040 [8, 6, 5, 5, 5, 1], 471, 120, 56520 [8, 6, 5, 5, 4, 2], 1009, 360, 363240 [8, 6, 5, 5, 3, 3], 1258, 180, 226440 [8, 6, 5, 4, 4, 3], 1566, 360, 563760 [8, 6, 4, 4, 4, 4], 1941, 30, 58230 [8, 5, 5, 5, 5, 2], 1233, 30, 36990 [8, 5, 5, 5, 4, 3], 1908, 120, 228960 [8, 5, 5, 4, 4, 4], 2358, 60, 141480 [7, 7, 7, 7, 2, 0], 15, 30, 450 [7, 7, 7, 7, 1, 1], 15, 15, 225 [7, 7, 7, 6, 3, 0], 51, 120, 6120 [7, 7, 7, 6, 2, 1], 102, 120, 12240 [7, 7, 7, 5, 4, 0], 87, 120, 10440 [7, 7, 7, 5, 3, 1], 231, 120, 27720 [7, 7, 7, 5, 2, 2], 318, 60, 19080 [7, 7, 7, 4, 4, 1], 303, 60, 18180 [7, 7, 7, 4, 3, 2], 519, 120, 62280 [7, 7, 7, 3, 3, 3], 648, 20, 12960 [7, 7, 6, 6, 4, 0], 111, 180, 19980 [7, 7, 6, 6, 3, 1], 295, 180, 53100 [7, 7, 6, 6, 2, 2], 406, 90, 36540 [7, 7, 6, 5, 5, 0], 138, 180, 24840 [7, 7, 6, 5, 4, 1], 476, 360, 171360 [7, 7, 6, 5, 3, 2], 814, 360, 293040 [7, 7, 6, 4, 4, 2], 1028, 180, 185040 [7, 7, 6, 4, 3, 3], 1282, 180, 230760 [7, 7, 5, 5, 5, 1], 588, 60, 35280 [7, 7, 5, 5, 4, 2], 1258, 180, 226440 [7, 7, 5, 5, 3, 3], 1570, 90, 141300 [7, 7, 5, 4, 4, 3], 1953, 180, 351540 [7, 7, 4, 4, 4, 4], 2418, 15, 36270 [7, 6, 6, 6, 5, 0], 174, 120, 20880 [7, 6, 6, 6, 4, 1], 597, 120, 71640 [7, 6, 6, 6, 3, 2], 1020, 120, 122400 [7, 6, 6, 5, 5, 1], 735, 180, 132300 [7, 6, 6, 5, 4, 2], 1566, 360, 563760 [7, 6, 6, 5, 3, 3], 1953, 180, 351540 [7, 6, 6, 4, 4, 3], 2424, 180, 436320 [7, 6, 5, 5, 5, 2], 1908, 120, 228960 [7, 6, 5, 5, 4, 3], 2943, 360, 1059480 [7, 6, 5, 4, 4, 4], 3627, 120, 435240 [7, 5, 5, 5, 5, 3], 3570, 30, 107100 [7, 5, 5, 5, 4, 4], 4389, 60, 263340 [6, 6, 6, 6, 6, 0], 219, 6, 1314 [6, 6, 6, 6, 5, 1], 915, 30, 27450 [6, 6, 6, 6, 4, 2], 1941, 30, 58230 [6, 6, 6, 6, 3, 3], 2418, 15, 36270 [6, 6, 6, 5, 5, 2], 2358, 60, 141480 [6, 6, 6, 5, 4, 3], 3627, 120, 435240 [6, 6, 6, 4, 4, 4], 4464, 20, 89280 [6, 6, 5, 5, 5, 3], 4389, 60, 263340 [6, 6, 5, 5, 4, 4], 5388, 90, 484920 [6, 5, 5, 5, 5, 4], 6501, 30, 195030 [5, 5, 5, 5, 5, 5], 7839, 1, 7839 Sum of numbers in last column=, 14348907 The most probable vector is, [7, 6, 5, 5, 4, 3], which arises in, 1059480, out of, 3, to the power, 15 that equals, 14348907, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 6, players The sole champion score-vector is, [7, 6, 5, 5, 4, 3] the corresponding record is, 1059480 ----------------------------------------------------------------------------\ ----- 7, Players. Symmetric Function:, 7 / 7 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [12, 10, 8, 6, 4, 2, 0], 1, 5040, 5040 [12, 10, 8, 6, 4, 1, 1], 1, 2520, 2520 [12, 10, 8, 6, 3, 3, 0], 1, 2520, 2520 [12, 10, 8, 6, 3, 2, 1], 2, 5040, 10080 [12, 10, 8, 6, 2, 2, 2], 3, 840, 2520 [12, 10, 8, 5, 5, 2, 0], 1, 2520, 2520 [12, 10, 8, 5, 5, 1, 1], 1, 1260, 1260 [12, 10, 8, 5, 4, 3, 0], 2, 5040, 10080 [12, 10, 8, 5, 4, 2, 1], 4, 5040, 20160 [12, 10, 8, 5, 3, 3, 1], 5, 2520, 12600 [12, 10, 8, 5, 3, 2, 2], 7, 2520, 17640 [12, 10, 8, 4, 4, 4, 0], 3, 840, 2520 [12, 10, 8, 4, 4, 3, 1], 7, 2520, 17640 [12, 10, 8, 4, 4, 2, 2], 10, 1260, 12600 [12, 10, 8, 4, 3, 3, 2], 12, 2520, 30240 [12, 10, 8, 3, 3, 3, 3], 15, 210, 3150 [12, 10, 7, 7, 4, 2, 0], 1, 2520, 2520 [12, 10, 7, 7, 4, 1, 1], 1, 1260, 1260 [12, 10, 7, 7, 3, 3, 0], 1, 1260, 1260 [12, 10, 7, 7, 3, 2, 1], 2, 2520, 5040 [12, 10, 7, 7, 2, 2, 2], 3, 420, 1260 [12, 10, 7, 6, 5, 2, 0], 2, 5040, 10080 [12, 10, 7, 6, 5, 1, 1], 2, 2520, 5040 [12, 10, 7, 6, 4, 3, 0], 4, 5040, 20160 [12, 10, 7, 6, 4, 2, 1], 8, 5040, 40320 [12, 10, 7, 6, 3, 3, 1], 10, 2520, 25200 [12, 10, 7, 6, 3, 2, 2], 14, 2520, 35280 [12, 10, 7, 5, 5, 3, 0], 5, 2520, 12600 [12, 10, 7, 5, 5, 2, 1], 10, 2520, 25200 [12, 10, 7, 5, 4, 4, 0], 7, 2520, 17640 [12, 10, 7, 5, 4, 3, 1], 18, 5040, 90720 [12, 10, 7, 5, 4, 2, 2], 25, 2520, 63000 [12, 10, 7, 5, 3, 3, 2], 31, 2520, 78120 [12, 10, 7, 4, 4, 4, 1], 24, 840, 20160 [12, 10, 7, 4, 4, 3, 2], 41, 2520, 103320 [12, 10, 7, 4, 3, 3, 3], 51, 840, 42840 [12, 10, 6, 6, 6, 2, 0], 3, 840, 2520 [12, 10, 6, 6, 6, 1, 1], 3, 420, 1260 [12, 10, 6, 6, 5, 3, 0], 7, 2520, 17640 [12, 10, 6, 6, 5, 2, 1], 14, 2520, 35280 [12, 10, 6, 6, 4, 4, 0], 10, 1260, 12600 [12, 10, 6, 6, 4, 3, 1], 25, 2520, 63000 [12, 10, 6, 6, 4, 2, 2], 35, 1260, 44100 [12, 10, 6, 6, 3, 3, 2], 43, 1260, 54180 [12, 10, 6, 5, 5, 4, 0], 12, 2520, 30240 [12, 10, 6, 5, 5, 3, 1], 31, 2520, 78120 [12, 10, 6, 5, 5, 2, 2], 43, 1260, 54180 [12, 10, 6, 5, 4, 4, 1], 41, 2520, 103320 [12, 10, 6, 5, 4, 3, 2], 70, 5040, 352800 [12, 10, 6, 5, 3, 3, 3], 87, 840, 73080 [12, 10, 6, 4, 4, 4, 2], 90, 840, 75600 [12, 10, 6, 4, 4, 3, 3], 111, 1260, 139860 [12, 10, 5, 5, 5, 5, 0], 15, 210, 3150 [12, 10, 5, 5, 5, 4, 1], 51, 840, 42840 [12, 10, 5, 5, 5, 3, 2], 87, 840, 73080 [12, 10, 5, 5, 4, 4, 2], 111, 1260, 139860 [12, 10, 5, 5, 4, 3, 3], 138, 1260, 173880 [12, 10, 5, 4, 4, 4, 3], 174, 840, 146160 [12, 10, 4, 4, 4, 4, 4], 219, 42, 9198 [12, 9, 9, 6, 4, 2, 0], 1, 2520, 2520 [12, 9, 9, 6, 4, 1, 1], 1, 1260, 1260 [12, 9, 9, 6, 3, 3, 0], 1, 1260, 1260 [12, 9, 9, 6, 3, 2, 1], 2, 2520, 5040 [12, 9, 9, 6, 2, 2, 2], 3, 420, 1260 [12, 9, 9, 5, 5, 2, 0], 1, 1260, 1260 [12, 9, 9, 5, 5, 1, 1], 1, 630, 630 [12, 9, 9, 5, 4, 3, 0], 2, 2520, 5040 [12, 9, 9, 5, 4, 2, 1], 4, 2520, 10080 [12, 9, 9, 5, 3, 3, 1], 5, 1260, 6300 [12, 9, 9, 5, 3, 2, 2], 7, 1260, 8820 [12, 9, 9, 4, 4, 4, 0], 3, 420, 1260 [12, 9, 9, 4, 4, 3, 1], 7, 1260, 8820 [12, 9, 9, 4, 4, 2, 2], 10, 630, 6300 [12, 9, 9, 4, 3, 3, 2], 12, 1260, 15120 [12, 9, 9, 3, 3, 3, 3], 15, 105, 1575 [12, 9, 8, 7, 4, 2, 0], 2, 5040, 10080 [12, 9, 8, 7, 4, 1, 1], 2, 2520, 5040 [12, 9, 8, 7, 3, 3, 0], 2, 2520, 5040 [12, 9, 8, 7, 3, 2, 1], 4, 5040, 20160 [12, 9, 8, 7, 2, 2, 2], 6, 840, 5040 [12, 9, 8, 6, 5, 2, 0], 4, 5040, 20160 [12, 9, 8, 6, 5, 1, 1], 4, 2520, 10080 [12, 9, 8, 6, 4, 3, 0], 8, 5040, 40320 [12, 9, 8, 6, 4, 2, 1], 16, 5040, 80640 [12, 9, 8, 6, 3, 3, 1], 20, 2520, 50400 [12, 9, 8, 6, 3, 2, 2], 28, 2520, 70560 [12, 9, 8, 5, 5, 3, 0], 10, 2520, 25200 [12, 9, 8, 5, 5, 2, 1], 20, 2520, 50400 [12, 9, 8, 5, 4, 4, 0], 14, 2520, 35280 [12, 9, 8, 5, 4, 3, 1], 36, 5040, 181440 [12, 9, 8, 5, 4, 2, 2], 50, 2520, 126000 [12, 9, 8, 5, 3, 3, 2], 62, 2520, 156240 [12, 9, 8, 4, 4, 4, 1], 48, 840, 40320 [12, 9, 8, 4, 4, 3, 2], 82, 2520, 206640 [12, 9, 8, 4, 3, 3, 3], 102, 840, 85680 [12, 9, 7, 7, 5, 2, 0], 5, 2520, 12600 [12, 9, 7, 7, 5, 1, 1], 5, 1260, 6300 [12, 9, 7, 7, 4, 3, 0], 10, 2520, 25200 [12, 9, 7, 7, 4, 2, 1], 20, 2520, 50400 [12, 9, 7, 7, 3, 3, 1], 25, 1260, 31500 [12, 9, 7, 7, 3, 2, 2], 35, 1260, 44100 [12, 9, 7, 6, 6, 2, 0], 7, 2520, 17640 [12, 9, 7, 6, 6, 1, 1], 7, 1260, 8820 [12, 9, 7, 6, 5, 3, 0], 18, 5040, 90720 [12, 9, 7, 6, 5, 2, 1], 36, 5040, 181440 [12, 9, 7, 6, 4, 4, 0], 25, 2520, 63000 [12, 9, 7, 6, 4, 3, 1], 65, 5040, 327600 [12, 9, 7, 6, 4, 2, 2], 90, 2520, 226800 [12, 9, 7, 6, 3, 3, 2], 112, 2520, 282240 [12, 9, 7, 5, 5, 4, 0], 31, 2520, 78120 [12, 9, 7, 5, 5, 3, 1], 82, 2520, 206640 [12, 9, 7, 5, 5, 2, 2], 113, 1260, 142380 [12, 9, 7, 5, 4, 4, 1], 108, 2520, 272160 [12, 9, 7, 5, 4, 3, 2], 185, 5040, 932400 [12, 9, 7, 5, 3, 3, 3], 231, 840, 194040 [12, 9, 7, 4, 4, 4, 2], 237, 840, 199080 [12, 9, 7, 4, 4, 3, 3], 295, 1260, 371700 [12, 9, 6, 6, 6, 3, 0], 24, 840, 20160 [12, 9, 6, 6, 6, 2, 1], 48, 840, 40320 [12, 9, 6, 6, 5, 4, 0], 41, 2520, 103320 [12, 9, 6, 6, 5, 3, 1], 108, 2520, 272160 [12, 9, 6, 6, 5, 2, 2], 149, 1260, 187740 [12, 9, 6, 6, 4, 4, 1], 142, 1260, 178920 [12, 9, 6, 6, 4, 3, 2], 243, 2520, 612360 [12, 9, 6, 6, 3, 3, 3], 303, 420, 127260 [12, 9, 6, 5, 5, 5, 0], 51, 840, 42840 [12, 9, 6, 5, 5, 4, 1], 176, 2520, 443520 [12, 9, 6, 5, 5, 3, 2], 301, 2520, 758520 [12, 9, 6, 5, 4, 4, 2], 382, 2520, 962640 [12, 9, 6, 5, 4, 3, 3], 476, 2520, 1199520 [12, 9, 6, 4, 4, 4, 3], 597, 840, 501480 [12, 9, 5, 5, 5, 5, 1], 219, 210, 45990 [12, 9, 5, 5, 5, 4, 2], 471, 840, 395640 [12, 9, 5, 5, 5, 3, 3], 588, 420, 246960 [12, 9, 5, 5, 4, 4, 3], 735, 1260, 926100 [12, 9, 5, 4, 4, 4, 4], 915, 210, 192150 [12, 8, 8, 8, 4, 2, 0], 3, 840, 2520 [12, 8, 8, 8, 4, 1, 1], 3, 420, 1260 [12, 8, 8, 8, 3, 3, 0], 3, 420, 1260 [12, 8, 8, 8, 3, 2, 1], 6, 840, 5040 [12, 8, 8, 8, 2, 2, 2], 9, 140, 1260 [12, 8, 8, 7, 5, 2, 0], 7, 2520, 17640 [12, 8, 8, 7, 5, 1, 1], 7, 1260, 8820 [12, 8, 8, 7, 4, 3, 0], 14, 2520, 35280 [12, 8, 8, 7, 4, 2, 1], 28, 2520, 70560 [12, 8, 8, 7, 3, 3, 1], 35, 1260, 44100 [12, 8, 8, 7, 3, 2, 2], 49, 1260, 61740 [12, 8, 8, 6, 6, 2, 0], 10, 1260, 12600 [12, 8, 8, 6, 6, 1, 1], 10, 630, 6300 [12, 8, 8, 6, 5, 3, 0], 25, 2520, 63000 [12, 8, 8, 6, 5, 2, 1], 50, 2520, 126000 [12, 8, 8, 6, 4, 4, 0], 35, 1260, 44100 [12, 8, 8, 6, 4, 3, 1], 90, 2520, 226800 [12, 8, 8, 6, 4, 2, 2], 125, 1260, 157500 [12, 8, 8, 6, 3, 3, 2], 155, 1260, 195300 [12, 8, 8, 5, 5, 4, 0], 43, 1260, 54180 [12, 8, 8, 5, 5, 3, 1], 113, 1260, 142380 [12, 8, 8, 5, 5, 2, 2], 156, 630, 98280 [12, 8, 8, 5, 4, 4, 1], 149, 1260, 187740 [12, 8, 8, 5, 4, 3, 2], 255, 2520, 642600 [12, 8, 8, 5, 3, 3, 3], 318, 420, 133560 [12, 8, 8, 4, 4, 4, 2], 327, 420, 137340 [12, 8, 8, 4, 4, 3, 3], 406, 630, 255780 [12, 8, 7, 7, 6, 2, 0], 12, 2520, 30240 [12, 8, 7, 7, 6, 1, 1], 12, 1260, 15120 [12, 8, 7, 7, 5, 3, 0], 31, 2520, 78120 [12, 8, 7, 7, 5, 2, 1], 62, 2520, 156240 [12, 8, 7, 7, 4, 4, 0], 43, 1260, 54180 [12, 8, 7, 7, 4, 3, 1], 112, 2520, 282240 [12, 8, 7, 7, 4, 2, 2], 155, 1260, 195300 [12, 8, 7, 7, 3, 3, 2], 193, 1260, 243180 [12, 8, 7, 6, 6, 3, 0], 41, 2520, 103320 [12, 8, 7, 6, 6, 2, 1], 82, 2520, 206640 [12, 8, 7, 6, 5, 4, 0], 70, 5040, 352800 [12, 8, 7, 6, 5, 3, 1], 185, 5040, 932400 [12, 8, 7, 6, 5, 2, 2], 255, 2520, 642600 [12, 8, 7, 6, 4, 4, 1], 243, 2520, 612360 [12, 8, 7, 6, 4, 3, 2], 416, 5040, 2096640 [12, 8, 7, 6, 3, 3, 3], 519, 840, 435960 [12, 8, 7, 5, 5, 5, 0], 87, 840, 73080 [12, 8, 7, 5, 5, 4, 1], 301, 2520, 758520 [12, 8, 7, 5, 5, 3, 2], 515, 2520, 1297800 [12, 8, 7, 5, 4, 4, 2], 653, 2520, 1645560 [12, 8, 7, 5, 4, 3, 3], 814, 2520, 2051280 [12, 8, 7, 4, 4, 4, 3], 1020, 840, 856800 [12, 8, 6, 6, 6, 4, 0], 90, 840, 75600 [12, 8, 6, 6, 6, 3, 1], 237, 840, 199080 [12, 8, 6, 6, 6, 2, 2], 327, 420, 137340 [12, 8, 6, 6, 5, 5, 0], 111, 1260, 139860 [12, 8, 6, 6, 5, 4, 1], 382, 2520, 962640 [12, 8, 6, 6, 5, 3, 2], 653, 2520, 1645560 [12, 8, 6, 6, 4, 4, 2], 826, 1260, 1040760 [12, 8, 6, 6, 4, 3, 3], 1028, 1260, 1295280 [12, 8, 6, 5, 5, 5, 1], 471, 840, 395640 [12, 8, 6, 5, 5, 4, 2], 1009, 2520, 2542680 [12, 8, 6, 5, 5, 3, 3], 1258, 1260, 1585080 [12, 8, 6, 5, 4, 4, 3], 1566, 2520, 3946320 [12, 8, 6, 4, 4, 4, 4], 1941, 210, 407610 [12, 8, 5, 5, 5, 5, 2], 1233, 210, 258930 [12, 8, 5, 5, 5, 4, 3], 1908, 840, 1602720 [12, 8, 5, 5, 4, 4, 4], 2358, 420, 990360 [12, 7, 7, 7, 7, 2, 0], 15, 210, 3150 [12, 7, 7, 7, 7, 1, 1], 15, 105, 1575 [12, 7, 7, 7, 6, 3, 0], 51, 840, 42840 [12, 7, 7, 7, 6, 2, 1], 102, 840, 85680 [12, 7, 7, 7, 5, 4, 0], 87, 840, 73080 [12, 7, 7, 7, 5, 3, 1], 231, 840, 194040 [12, 7, 7, 7, 5, 2, 2], 318, 420, 133560 [12, 7, 7, 7, 4, 4, 1], 303, 420, 127260 [12, 7, 7, 7, 4, 3, 2], 519, 840, 435960 [12, 7, 7, 7, 3, 3, 3], 648, 140, 90720 [12, 7, 7, 6, 6, 4, 0], 111, 1260, 139860 [12, 7, 7, 6, 6, 3, 1], 295, 1260, 371700 [12, 7, 7, 6, 6, 2, 2], 406, 630, 255780 [12, 7, 7, 6, 5, 5, 0], 138, 1260, 173880 [12, 7, 7, 6, 5, 4, 1], 476, 2520, 1199520 [12, 7, 7, 6, 5, 3, 2], 814, 2520, 2051280 [12, 7, 7, 6, 4, 4, 2], 1028, 1260, 1295280 [12, 7, 7, 6, 4, 3, 3], 1282, 1260, 1615320 [12, 7, 7, 5, 5, 5, 1], 588, 420, 246960 [12, 7, 7, 5, 5, 4, 2], 1258, 1260, 1585080 [12, 7, 7, 5, 5, 3, 3], 1570, 630, 989100 [12, 7, 7, 5, 4, 4, 3], 1953, 1260, 2460780 [12, 7, 7, 4, 4, 4, 4], 2418, 105, 253890 [12, 7, 6, 6, 6, 5, 0], 174, 840, 146160 [12, 7, 6, 6, 6, 4, 1], 597, 840, 501480 [12, 7, 6, 6, 6, 3, 2], 1020, 840, 856800 [12, 7, 6, 6, 5, 5, 1], 735, 1260, 926100 [12, 7, 6, 6, 5, 4, 2], 1566, 2520, 3946320 [12, 7, 6, 6, 5, 3, 3], 1953, 1260, 2460780 [12, 7, 6, 6, 4, 4, 3], 2424, 1260, 3054240 [12, 7, 6, 5, 5, 5, 2], 1908, 840, 1602720 [12, 7, 6, 5, 5, 4, 3], 2943, 2520, 7416360 [12, 7, 6, 5, 4, 4, 4], 3627, 840, 3046680 [12, 7, 5, 5, 5, 5, 3], 3570, 210, 749700 [12, 7, 5, 5, 5, 4, 4], 4389, 420, 1843380 [12, 6, 6, 6, 6, 6, 0], 219, 42, 9198 [12, 6, 6, 6, 6, 5, 1], 915, 210, 192150 [12, 6, 6, 6, 6, 4, 2], 1941, 210, 407610 [12, 6, 6, 6, 6, 3, 3], 2418, 105, 253890 [12, 6, 6, 6, 5, 5, 2], 2358, 420, 990360 [12, 6, 6, 6, 5, 4, 3], 3627, 840, 3046680 [12, 6, 6, 6, 4, 4, 4], 4464, 140, 624960 [12, 6, 6, 5, 5, 5, 3], 4389, 420, 1843380 [12, 6, 6, 5, 5, 4, 4], 5388, 630, 3394440 [12, 6, 5, 5, 5, 5, 4], 6501, 210, 1365210 [12, 5, 5, 5, 5, 5, 5], 7839, 7, 54873 [11, 11, 8, 6, 4, 2, 0], 1, 2520, 2520 [11, 11, 8, 6, 4, 1, 1], 1, 1260, 1260 [11, 11, 8, 6, 3, 3, 0], 1, 1260, 1260 [11, 11, 8, 6, 3, 2, 1], 2, 2520, 5040 [11, 11, 8, 6, 2, 2, 2], 3, 420, 1260 [11, 11, 8, 5, 5, 2, 0], 1, 1260, 1260 [11, 11, 8, 5, 5, 1, 1], 1, 630, 630 [11, 11, 8, 5, 4, 3, 0], 2, 2520, 5040 [11, 11, 8, 5, 4, 2, 1], 4, 2520, 10080 [11, 11, 8, 5, 3, 3, 1], 5, 1260, 6300 [11, 11, 8, 5, 3, 2, 2], 7, 1260, 8820 [11, 11, 8, 4, 4, 4, 0], 3, 420, 1260 [11, 11, 8, 4, 4, 3, 1], 7, 1260, 8820 [11, 11, 8, 4, 4, 2, 2], 10, 630, 6300 [11, 11, 8, 4, 3, 3, 2], 12, 1260, 15120 [11, 11, 8, 3, 3, 3, 3], 15, 105, 1575 [11, 11, 7, 7, 4, 2, 0], 1, 1260, 1260 [11, 11, 7, 7, 4, 1, 1], 1, 630, 630 [11, 11, 7, 7, 3, 3, 0], 1, 630, 630 [11, 11, 7, 7, 3, 2, 1], 2, 1260, 2520 [11, 11, 7, 7, 2, 2, 2], 3, 210, 630 [11, 11, 7, 6, 5, 2, 0], 2, 2520, 5040 [11, 11, 7, 6, 5, 1, 1], 2, 1260, 2520 [11, 11, 7, 6, 4, 3, 0], 4, 2520, 10080 [11, 11, 7, 6, 4, 2, 1], 8, 2520, 20160 [11, 11, 7, 6, 3, 3, 1], 10, 1260, 12600 [11, 11, 7, 6, 3, 2, 2], 14, 1260, 17640 [11, 11, 7, 5, 5, 3, 0], 5, 1260, 6300 [11, 11, 7, 5, 5, 2, 1], 10, 1260, 12600 [11, 11, 7, 5, 4, 4, 0], 7, 1260, 8820 [11, 11, 7, 5, 4, 3, 1], 18, 2520, 45360 [11, 11, 7, 5, 4, 2, 2], 25, 1260, 31500 [11, 11, 7, 5, 3, 3, 2], 31, 1260, 39060 [11, 11, 7, 4, 4, 4, 1], 24, 420, 10080 [11, 11, 7, 4, 4, 3, 2], 41, 1260, 51660 [11, 11, 7, 4, 3, 3, 3], 51, 420, 21420 [11, 11, 6, 6, 6, 2, 0], 3, 420, 1260 [11, 11, 6, 6, 6, 1, 1], 3, 210, 630 [11, 11, 6, 6, 5, 3, 0], 7, 1260, 8820 [11, 11, 6, 6, 5, 2, 1], 14, 1260, 17640 [11, 11, 6, 6, 4, 4, 0], 10, 630, 6300 [11, 11, 6, 6, 4, 3, 1], 25, 1260, 31500 [11, 11, 6, 6, 4, 2, 2], 35, 630, 22050 [11, 11, 6, 6, 3, 3, 2], 43, 630, 27090 [11, 11, 6, 5, 5, 4, 0], 12, 1260, 15120 [11, 11, 6, 5, 5, 3, 1], 31, 1260, 39060 [11, 11, 6, 5, 5, 2, 2], 43, 630, 27090 [11, 11, 6, 5, 4, 4, 1], 41, 1260, 51660 [11, 11, 6, 5, 4, 3, 2], 70, 2520, 176400 [11, 11, 6, 5, 3, 3, 3], 87, 420, 36540 [11, 11, 6, 4, 4, 4, 2], 90, 420, 37800 [11, 11, 6, 4, 4, 3, 3], 111, 630, 69930 [11, 11, 5, 5, 5, 5, 0], 15, 105, 1575 [11, 11, 5, 5, 5, 4, 1], 51, 420, 21420 [11, 11, 5, 5, 5, 3, 2], 87, 420, 36540 [11, 11, 5, 5, 4, 4, 2], 111, 630, 69930 [11, 11, 5, 5, 4, 3, 3], 138, 630, 86940 [11, 11, 5, 4, 4, 4, 3], 174, 420, 73080 [11, 11, 4, 4, 4, 4, 4], 219, 21, 4599 [11, 10, 9, 6, 4, 2, 0], 2, 5040, 10080 [11, 10, 9, 6, 4, 1, 1], 2, 2520, 5040 [11, 10, 9, 6, 3, 3, 0], 2, 2520, 5040 [11, 10, 9, 6, 3, 2, 1], 4, 5040, 20160 [11, 10, 9, 6, 2, 2, 2], 6, 840, 5040 [11, 10, 9, 5, 5, 2, 0], 2, 2520, 5040 [11, 10, 9, 5, 5, 1, 1], 2, 1260, 2520 [11, 10, 9, 5, 4, 3, 0], 4, 5040, 20160 [11, 10, 9, 5, 4, 2, 1], 8, 5040, 40320 [11, 10, 9, 5, 3, 3, 1], 10, 2520, 25200 [11, 10, 9, 5, 3, 2, 2], 14, 2520, 35280 [11, 10, 9, 4, 4, 4, 0], 6, 840, 5040 [11, 10, 9, 4, 4, 3, 1], 14, 2520, 35280 [11, 10, 9, 4, 4, 2, 2], 20, 1260, 25200 [11, 10, 9, 4, 3, 3, 2], 24, 2520, 60480 [11, 10, 9, 3, 3, 3, 3], 30, 210, 6300 [11, 10, 8, 7, 4, 2, 0], 4, 5040, 20160 [11, 10, 8, 7, 4, 1, 1], 4, 2520, 10080 [11, 10, 8, 7, 3, 3, 0], 4, 2520, 10080 [11, 10, 8, 7, 3, 2, 1], 8, 5040, 40320 [11, 10, 8, 7, 2, 2, 2], 12, 840, 10080 [11, 10, 8, 6, 5, 2, 0], 8, 5040, 40320 [11, 10, 8, 6, 5, 1, 1], 8, 2520, 20160 [11, 10, 8, 6, 4, 3, 0], 16, 5040, 80640 [11, 10, 8, 6, 4, 2, 1], 32, 5040, 161280 [11, 10, 8, 6, 3, 3, 1], 40, 2520, 100800 [11, 10, 8, 6, 3, 2, 2], 56, 2520, 141120 [11, 10, 8, 5, 5, 3, 0], 20, 2520, 50400 [11, 10, 8, 5, 5, 2, 1], 40, 2520, 100800 [11, 10, 8, 5, 4, 4, 0], 28, 2520, 70560 [11, 10, 8, 5, 4, 3, 1], 72, 5040, 362880 [11, 10, 8, 5, 4, 2, 2], 100, 2520, 252000 [11, 10, 8, 5, 3, 3, 2], 124, 2520, 312480 [11, 10, 8, 4, 4, 4, 1], 96, 840, 80640 [11, 10, 8, 4, 4, 3, 2], 164, 2520, 413280 [11, 10, 8, 4, 3, 3, 3], 204, 840, 171360 [11, 10, 7, 7, 5, 2, 0], 10, 2520, 25200 [11, 10, 7, 7, 5, 1, 1], 10, 1260, 12600 [11, 10, 7, 7, 4, 3, 0], 20, 2520, 50400 [11, 10, 7, 7, 4, 2, 1], 40, 2520, 100800 [11, 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5, 1], 5532, 420, 2323440 [9, 9, 8, 5, 5, 4, 2], 11848, 1260, 14928480 [9, 9, 8, 5, 5, 3, 3], 14816, 630, 9334080 [9, 9, 8, 5, 4, 4, 3], 18407, 1260, 23192820 [9, 9, 8, 4, 4, 4, 4], 22761, 105, 2389905 [9, 9, 7, 7, 7, 3, 0], 588, 420, 246960 [9, 9, 7, 7, 7, 2, 1], 1176, 420, 493920 [9, 9, 7, 7, 6, 4, 0], 1258, 1260, 1585080 [9, 9, 7, 7, 6, 3, 1], 3400, 1260, 4284000 [9, 9, 7, 7, 6, 2, 2], 4658, 630, 2934540 [9, 9, 7, 7, 5, 5, 0], 1570, 630, 989100 [9, 9, 7, 7, 5, 4, 1], 5486, 1260, 6912360 [9, 9, 7, 7, 5, 3, 2], 9402, 1260, 11846520 [9, 9, 7, 7, 4, 4, 2], 11846, 630, 7462980 [9, 9, 7, 7, 4, 3, 3], 14816, 630, 9334080 [9, 9, 7, 6, 6, 5, 0], 1953, 1260, 2460780 [9, 9, 7, 6, 6, 4, 1], 6803, 1260, 8571780 [9, 9, 7, 6, 6, 3, 2], 11653, 1260, 14682780 [9, 9, 7, 6, 5, 5, 1], 8386, 1260, 10566360 [9, 9, 7, 6, 5, 4, 2], 17870, 2520, 45032400 [9, 9, 7, 6, 5, 3, 3], 22350, 1260, 28161000 [9, 9, 7, 6, 4, 4, 3], 27679, 1260, 34875540 [9, 9, 7, 5, 5, 5, 2], 21765, 420, 9141300 [9, 9, 7, 5, 5, 4, 3], 33574, 1260, 42303240 [9, 9, 7, 5, 4, 4, 4], 41286, 420, 17340120 [9, 9, 6, 6, 6, 6, 0], 2418, 105, 253890 [9, 9, 6, 6, 6, 5, 1], 10317, 420, 4333140 [9, 9, 6, 6, 6, 4, 2], 21894, 420, 9195480 [9, 9, 6, 6, 6, 3, 3], 27375, 210, 5748750 [9, 9, 6, 6, 5, 5, 2], 26602, 630, 16759260 [9, 9, 6, 6, 5, 4, 3], 40934, 1260, 51576840 [9, 9, 6, 6, 4, 4, 4], 50250, 210, 10552500 [9, 9, 6, 5, 5, 5, 3], 49449, 420, 20768580 [9, 9, 6, 5, 5, 4, 4], 60561, 630, 38153430 [9, 9, 5, 5, 5, 5, 4], 72897, 105, 7654185 [9, 8, 8, 8, 7, 2, 0], 174, 840, 146160 [9, 8, 8, 8, 7, 1, 1], 174, 420, 73080 [9, 8, 8, 8, 6, 3, 0], 597, 840, 501480 [9, 8, 8, 8, 6, 2, 1], 1194, 840, 1002960 [9, 8, 8, 8, 5, 4, 0], 1020, 840, 856800 [9, 8, 8, 8, 5, 3, 1], 2739, 840, 2300760 [9, 8, 8, 8, 5, 2, 2], 3759, 420, 1578780 [9, 8, 8, 8, 4, 4, 1], 3585, 420, 1505700 [9, 8, 8, 8, 4, 3, 2], 6150, 840, 5166000 [9, 8, 8, 8, 3, 3, 3], 7695, 140, 1077300 [9, 8, 8, 7, 7, 3, 0], 735, 1260, 926100 [9, 8, 8, 7, 7, 2, 1], 1470, 1260, 1852200 [9, 8, 8, 7, 6, 4, 0], 1566, 2520, 3946320 [9, 8, 8, 7, 6, 3, 1], 4232, 2520, 10664640 [9, 8, 8, 7, 6, 2, 2], 5798, 1260, 7305480 [9, 8, 8, 7, 5, 5, 0], 1953, 1260, 2460780 [9, 8, 8, 7, 5, 4, 1], 6820, 2520, 17186400 [9, 8, 8, 7, 5, 3, 2], 11687, 2520, 29451240 [9, 8, 8, 7, 4, 4, 2], 14719, 1260, 18545940 [9, 8, 8, 7, 4, 3, 3], 18407, 1260, 23192820 [9, 8, 8, 6, 6, 5, 0], 2424, 1260, 3054240 [9, 8, 8, 6, 6, 4, 1], 8438, 1260, 10631880 [9, 8, 8, 6, 6, 3, 2], 14452, 1260, 18209520 [9, 8, 8, 6, 5, 5, 1], 10393, 1260, 13095180 [9, 8, 8, 6, 5, 4, 2], 22134, 2520, 55777680 [9, 8, 8, 6, 5, 3, 3], 27679, 1260, 34875540 [9, 8, 8, 6, 4, 4, 3], 34264, 1260, 43172640 [9, 8, 8, 5, 5, 5, 2], 26934, 420, 11312280 [9, 8, 8, 5, 5, 4, 3], 41522, 1260, 52317720 [9, 8, 8, 5, 4, 4, 4], 51036, 420, 21435120 [9, 8, 7, 7, 7, 4, 0], 1908, 840, 1602720 [9, 8, 7, 7, 7, 3, 1], 5172, 840, 4344480 [9, 8, 7, 7, 7, 2, 2], 7080, 420, 2973600 [9, 8, 7, 7, 6, 5, 0], 2943, 2520, 7416360 [9, 8, 7, 7, 6, 4, 1], 10250, 2520, 25830000 [9, 8, 7, 7, 6, 3, 2], 17557, 2520, 44243640 [9, 8, 7, 7, 5, 5, 1], 12620, 1260, 15901200 [9, 8, 7, 7, 5, 4, 2], 26840, 2520, 67636800 [9, 8, 7, 7, 5, 3, 3], 33574, 1260, 42303240 [9, 8, 7, 7, 4, 4, 3], 41522, 1260, 52317720 [9, 8, 7, 6, 6, 6, 0], 3627, 840, 3046680 [9, 8, 7, 6, 6, 5, 1], 15456, 2520, 38949120 [9, 8, 7, 6, 6, 4, 2], 32732, 2520, 82484640 [9, 8, 7, 6, 6, 3, 3], 40934, 1260, 51576840 [9, 8, 7, 6, 5, 5, 2], 39715, 2520, 100081800 [9, 8, 7, 6, 5, 4, 3], 61031, 5040, 307596240 [9, 8, 7, 6, 4, 4, 4], 74829, 840, 62856360 [9, 8, 7, 5, 5, 5, 3], 73602, 840, 61825680 [9, 8, 7, 5, 5, 4, 4], 90030, 1260, 113437800 [9, 8, 6, 6, 6, 6, 1], 18867, 210, 3962070 [9, 8, 6, 6, 6, 5, 2], 48162, 840, 40456080 [9, 8, 6, 6, 6, 4, 3], 73830, 840, 62017200 [9, 8, 6, 6, 5, 5, 3], 88872, 1260, 111978720 [9, 8, 6, 6, 5, 4, 4], 108546, 1260, 136767960 [9, 8, 6, 5, 5, 5, 4], 130200, 840, 109368000 [9, 8, 5, 5, 5, 5, 5], 155985, 42, 6551370 [9, 7, 7, 7, 7, 5, 0], 3570, 210, 749700 [9, 7, 7, 7, 7, 4, 1], 12435, 210, 2611350 [9, 7, 7, 7, 7, 3, 2], 21300, 210, 4473000 [9, 7, 7, 7, 6, 6, 0], 4389, 420, 1843380 [9, 7, 7, 7, 6, 5, 1], 18696, 840, 15704640 [9, 7, 7, 7, 6, 4, 2], 39531, 840, 33206040 [9, 7, 7, 7, 6, 3, 3], 49449, 420, 20768580 [9, 7, 7, 7, 5, 5, 2], 47934, 420, 20132280 [9, 7, 7, 7, 5, 4, 3], 73602, 840, 61825680 [9, 7, 7, 7, 4, 4, 4], 90171, 140, 12623940 [9, 7, 7, 6, 6, 6, 1], 22779, 420, 9567180 [9, 7, 7, 6, 6, 5, 2], 58020, 1260, 73105200 [9, 7, 7, 6, 6, 4, 3], 88872, 1260, 111978720 [9, 7, 7, 6, 5, 5, 3], 106908, 1260, 134704080 [9, 7, 7, 6, 5, 4, 4], 130479, 1260, 164403540 [9, 7, 7, 5, 5, 5, 4], 156402, 420, 65688840 [9, 7, 6, 6, 6, 6, 2], 70056, 210, 14711760 [9, 7, 6, 6, 6, 5, 3], 128559, 840, 107989560 [9, 7, 6, 6, 6, 4, 4], 156684, 420, 65807280 [9, 7, 6, 6, 5, 5, 4], 187548, 1260, 236310480 [9, 7, 6, 5, 5, 5, 5], 224226, 210, 47087460 [9, 6, 6, 6, 6, 6, 3], 154290, 42, 6480180 [9, 6, 6, 6, 6, 5, 4], 224493, 210, 47143530 [9, 6, 6, 6, 5, 5, 5], 268065, 140, 37529100 [8, 8, 8, 8, 8, 2, 0], 219, 42, 9198 [8, 8, 8, 8, 8, 1, 1], 219, 21, 4599 [8, 8, 8, 8, 7, 3, 0], 915, 210, 192150 [8, 8, 8, 8, 7, 2, 1], 1830, 210, 384300 [8, 8, 8, 8, 6, 4, 0], 1941, 210, 407610 [8, 8, 8, 8, 6, 3, 1], 5244, 210, 1101240 [8, 8, 8, 8, 6, 2, 2], 7185, 105, 754425 [8, 8, 8, 8, 5, 5, 0], 2418, 105, 253890 [8, 8, 8, 8, 5, 4, 1], 8439, 210, 1772190 [8, 8, 8, 8, 5, 3, 2], 14460, 210, 3036600 [8, 8, 8, 8, 4, 4, 2], 18204, 105, 1911420 [8, 8, 8, 8, 4, 3, 3], 22761, 105, 2389905 [8, 8, 8, 7, 7, 4, 0], 2358, 420, 990360 [8, 8, 8, 7, 7, 3, 1], 6393, 420, 2685060 [8, 8, 8, 7, 7, 2, 2], 8751, 210, 1837710 [8, 8, 8, 7, 6, 5, 0], 3627, 840, 3046680 [8, 8, 8, 7, 6, 4, 1], 12624, 840, 10604160 [8, 8, 8, 7, 6, 3, 2], 21621, 840, 18161640 [8, 8, 8, 7, 5, 5, 1], 15531, 420, 6523020 [8, 8, 8, 7, 5, 4, 2], 33009, 840, 27727560 [8, 8, 8, 7, 5, 3, 3], 41286, 420, 17340120 [8, 8, 8, 7, 4, 4, 3], 51036, 420, 21435120 [8, 8, 8, 6, 6, 6, 0], 4464, 140, 624960 [8, 8, 8, 6, 6, 5, 1], 18990, 420, 7975800 [8, 8, 8, 6, 6, 4, 2], 40188, 420, 16878960 [8, 8, 8, 6, 6, 3, 3], 50250, 210, 10552500 [8, 8, 8, 6, 5, 5, 2], 48723, 420, 20463660 [8, 8, 8, 6, 5, 4, 3], 74829, 840, 62856360 [8, 8, 8, 6, 4, 4, 4], 91710, 140, 12839400 [8, 8, 8, 5, 5, 5, 3], 90171, 140, 12623940 [8, 8, 8, 5, 5, 4, 4], 110250, 210, 23152500 [8, 8, 7, 7, 7, 5, 0], 4389, 420, 1843380 [8, 8, 7, 7, 7, 4, 1], 15279, 420, 6417180 [8, 8, 7, 7, 7, 3, 2], 26169, 420, 10990980 [8, 8, 7, 7, 6, 6, 0], 5388, 630, 3394440 [8, 8, 7, 7, 6, 5, 1], 22917, 1260, 28875420 [8, 8, 7, 7, 6, 4, 2], 48420, 1260, 61009200 [8, 8, 7, 7, 6, 3, 3], 60561, 630, 38153430 [8, 8, 7, 7, 5, 5, 2], 58668, 630, 36960840 [8, 8, 7, 7, 5, 4, 3], 90030, 1260, 113437800 [8, 8, 7, 7, 4, 4, 4], 110250, 210, 23152500 [8, 8, 7, 6, 6, 6, 1], 27882, 420, 11710440 [8, 8, 7, 6, 6, 5, 2], 70908, 1260, 89344080 [8, 8, 7, 6, 6, 4, 3], 108546, 1260, 136767960 [8, 8, 7, 6, 5, 5, 3], 130479, 1260, 164403540 [8, 8, 7, 6, 5, 4, 4], 159180, 1260, 200566800 [8, 8, 7, 5, 5, 5, 4], 190665, 420, 80079300 [8, 8, 6, 6, 6, 6, 2], 85500, 105, 8977500 [8, 8, 6, 6, 6, 5, 3], 156684, 420, 65807280 [8, 8, 6, 6, 6, 4, 4], 190884, 210, 40085640 [8, 8, 6, 6, 5, 5, 4], 228318, 630, 143840340 [8, 8, 6, 5, 5, 5, 5], 272772, 105, 28641060 [8, 7, 7, 7, 7, 6, 0], 6501, 210, 1365210 [8, 7, 7, 7, 7, 5, 1], 27627, 210, 5801670 [8, 7, 7, 7, 7, 4, 2], 58272, 210, 12237120 [8, 7, 7, 7, 7, 3, 3], 72897, 105, 7654185 [8, 7, 7, 7, 6, 6, 1], 33555, 420, 14093100 [8, 7, 7, 7, 6, 5, 2], 85128, 840, 71507520 [8, 7, 7, 7, 6, 4, 3], 130200, 840, 109368000 [8, 7, 7, 7, 5, 5, 3], 156402, 420, 65688840 [8, 7, 7, 7, 5, 4, 4], 190665, 420, 80079300 [8, 7, 7, 6, 6, 6, 2], 102492, 420, 43046640 [8, 7, 7, 6, 6, 5, 3], 187548, 1260, 236310480 [8, 7, 7, 6, 6, 4, 4], 228318, 630, 143840340 [8, 7, 7, 6, 5, 5, 4], 272937, 1260, 343900620 [8, 7, 7, 5, 5, 5, 5], 325881, 105, 34217505 [8, 7, 6, 6, 6, 6, 3], 224493, 210, 47143530 [8, 7, 6, 6, 6, 5, 4], 325866, 840, 273727440 [8, 7, 6, 6, 5, 5, 5], 388620, 420, 163220400 [8, 6, 6, 6, 6, 6, 4], 388500, 42, 16317000 [8, 6, 6, 6, 6, 5, 5], 462801, 105, 48594105 [7, 7, 7, 7, 7, 7, 0], 7839, 7, 54873 [7, 7, 7, 7, 7, 6, 1], 40344, 42, 1694448 [7, 7, 7, 7, 7, 5, 2], 102084, 42, 4287528 [7, 7, 7, 7, 7, 4, 3], 155985, 42, 6551370 [7, 7, 7, 7, 6, 6, 2], 122733, 105, 12886965 [7, 7, 7, 7, 6, 5, 3], 224226, 210, 47087460 [7, 7, 7, 7, 6, 4, 4], 272772, 105, 28641060 [7, 7, 7, 7, 5, 5, 4], 325881, 105, 34217505 [7, 7, 7, 6, 6, 6, 3], 268065, 140, 37529100 [7, 7, 7, 6, 6, 5, 4], 388620, 420, 163220400 [7, 7, 7, 6, 5, 5, 5], 463194, 140, 64847160 [7, 7, 6, 6, 6, 6, 4], 462801, 105, 48594105 [7, 7, 6, 6, 6, 5, 5], 551019, 210, 115713990 [7, 6, 6, 6, 6, 6, 5], 654714, 42, 27497988 [6, 6, 6, 6, 6, 6, 6], 777069, 1, 777069 Sum of numbers in last column=, 10460353203 The most probable vector is, [8, 7, 7, 6, 5, 5, 4], which arises in, 343900620, out of, 3, to the power, 21 that equals, 10460353203, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 7, players The sole champion score-vector is, [8, 7, 7, 6, 5, 5, 4] the corresponding record is, 343900620 ----------------------------------------------------------------------------\ ----- 8, Players. Symmetric Function:, 8 / 8 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [14, 12, 10, 8, 6, 4, 2, 0], 1, 40320, 40320 [14, 12, 10, 8, 6, 4, 1, 1], 1, 20160, 20160 [14, 12, 10, 8, 6, 3, 3, 0], 1, 20160, 20160 [14, 12, 10, 8, 6, 3, 2, 1], 2, 40320, 80640 [14, 12, 10, 8, 6, 2, 2, 2], 3, 6720, 20160 [14, 12, 10, 8, 5, 5, 2, 0], 1, 20160, 20160 [14, 12, 10, 8, 5, 5, 1, 1], 1, 10080, 10080 [14, 12, 10, 8, 5, 4, 3, 0], 2, 40320, 80640 [14, 12, 10, 8, 5, 4, 2, 1], 4, 40320, 161280 [14, 12, 10, 8, 5, 3, 3, 1], 5, 20160, 100800 [14, 12, 10, 8, 5, 3, 2, 2], 7, 20160, 141120 [14, 12, 10, 8, 4, 4, 4, 0], 3, 6720, 20160 [14, 12, 10, 8, 4, 4, 3, 1], 7, 20160, 141120 [14, 12, 10, 8, 4, 4, 2, 2], 10, 10080, 100800 [14, 12, 10, 8, 4, 3, 3, 2], 12, 20160, 241920 [14, 12, 10, 8, 3, 3, 3, 3], 15, 1680, 25200 [14, 12, 10, 7, 7, 4, 2, 0], 1, 20160, 20160 [14, 12, 10, 7, 7, 4, 1, 1], 1, 10080, 10080 [14, 12, 10, 7, 7, 3, 3, 0], 1, 10080, 10080 [14, 12, 10, 7, 7, 3, 2, 1], 2, 20160, 40320 [14, 12, 10, 7, 7, 2, 2, 2], 3, 3360, 10080 [14, 12, 10, 7, 6, 5, 2, 0], 2, 40320, 80640 [14, 12, 10, 7, 6, 5, 1, 1], 2, 20160, 40320 [14, 12, 10, 7, 6, 4, 3, 0], 4, 40320, 161280 [14, 12, 10, 7, 6, 4, 2, 1], 8, 40320, 322560 [14, 12, 10, 7, 6, 3, 3, 1], 10, 20160, 201600 [14, 12, 10, 7, 6, 3, 2, 2], 14, 20160, 282240 [14, 12, 10, 7, 5, 5, 3, 0], 5, 20160, 100800 [14, 12, 10, 7, 5, 5, 2, 1], 10, 20160, 201600 [14, 12, 10, 7, 5, 4, 4, 0], 7, 20160, 141120 [14, 12, 10, 7, 5, 4, 3, 1], 18, 40320, 725760 [14, 12, 10, 7, 5, 4, 2, 2], 25, 20160, 504000 [14, 12, 10, 7, 5, 3, 3, 2], 31, 20160, 624960 [14, 12, 10, 7, 4, 4, 4, 1], 24, 6720, 161280 [14, 12, 10, 7, 4, 4, 3, 2], 41, 20160, 826560 [14, 12, 10, 7, 4, 3, 3, 3], 51, 6720, 342720 [14, 12, 10, 6, 6, 6, 2, 0], 3, 6720, 20160 [14, 12, 10, 6, 6, 6, 1, 1], 3, 3360, 10080 [14, 12, 10, 6, 6, 5, 3, 0], 7, 20160, 141120 [14, 12, 10, 6, 6, 5, 2, 1], 14, 20160, 282240 [14, 12, 10, 6, 6, 4, 4, 0], 10, 10080, 100800 [14, 12, 10, 6, 6, 4, 3, 1], 25, 20160, 504000 [14, 12, 10, 6, 6, 4, 2, 2], 35, 10080, 352800 [14, 12, 10, 6, 6, 3, 3, 2], 43, 10080, 433440 [14, 12, 10, 6, 5, 5, 4, 0], 12, 20160, 241920 [14, 12, 10, 6, 5, 5, 3, 1], 31, 20160, 624960 [14, 12, 10, 6, 5, 5, 2, 2], 43, 10080, 433440 [14, 12, 10, 6, 5, 4, 4, 1], 41, 20160, 826560 [14, 12, 10, 6, 5, 4, 3, 2], 70, 40320, 2822400 [14, 12, 10, 6, 5, 3, 3, 3], 87, 6720, 584640 [14, 12, 10, 6, 4, 4, 4, 2], 90, 6720, 604800 [14, 12, 10, 6, 4, 4, 3, 3], 111, 10080, 1118880 [14, 12, 10, 5, 5, 5, 5, 0], 15, 1680, 25200 [14, 12, 10, 5, 5, 5, 4, 1], 51, 6720, 342720 [14, 12, 10, 5, 5, 5, 3, 2], 87, 6720, 584640 [14, 12, 10, 5, 5, 4, 4, 2], 111, 10080, 1118880 [14, 12, 10, 5, 5, 4, 3, 3], 138, 10080, 1391040 [14, 12, 10, 5, 4, 4, 4, 3], 174, 6720, 1169280 [14, 12, 10, 4, 4, 4, 4, 4], 219, 336, 73584 [14, 12, 9, 9, 6, 4, 2, 0], 1, 20160, 20160 [14, 12, 9, 9, 6, 4, 1, 1], 1, 10080, 10080 [14, 12, 9, 9, 6, 3, 3, 0], 1, 10080, 10080 [14, 12, 9, 9, 6, 3, 2, 1], 2, 20160, 40320 [14, 12, 9, 9, 6, 2, 2, 2], 3, 3360, 10080 [14, 12, 9, 9, 5, 5, 2, 0], 1, 10080, 10080 [14, 12, 9, 9, 5, 5, 1, 1], 1, 5040, 5040 [14, 12, 9, 9, 5, 4, 3, 0], 2, 20160, 40320 [14, 12, 9, 9, 5, 4, 2, 1], 4, 20160, 80640 [14, 12, 9, 9, 5, 3, 3, 1], 5, 10080, 50400 [14, 12, 9, 9, 5, 3, 2, 2], 7, 10080, 70560 [14, 12, 9, 9, 4, 4, 4, 0], 3, 3360, 10080 [14, 12, 9, 9, 4, 4, 3, 1], 7, 10080, 70560 [14, 12, 9, 9, 4, 4, 2, 2], 10, 5040, 50400 [14, 12, 9, 9, 4, 3, 3, 2], 12, 10080, 120960 [14, 12, 9, 9, 3, 3, 3, 3], 15, 840, 12600 [14, 12, 9, 8, 7, 4, 2, 0], 2, 40320, 80640 [14, 12, 9, 8, 7, 4, 1, 1], 2, 20160, 40320 [14, 12, 9, 8, 7, 3, 3, 0], 2, 20160, 40320 [14, 12, 9, 8, 7, 3, 2, 1], 4, 40320, 161280 [14, 12, 9, 8, 7, 2, 2, 2], 6, 6720, 40320 [14, 12, 9, 8, 6, 5, 2, 0], 4, 40320, 161280 [14, 12, 9, 8, 6, 5, 1, 1], 4, 20160, 80640 [14, 12, 9, 8, 6, 4, 3, 0], 8, 40320, 322560 [14, 12, 9, 8, 6, 4, 2, 1], 16, 40320, 645120 [14, 12, 9, 8, 6, 3, 3, 1], 20, 20160, 403200 [14, 12, 9, 8, 6, 3, 2, 2], 28, 20160, 564480 [14, 12, 9, 8, 5, 5, 3, 0], 10, 20160, 201600 [14, 12, 9, 8, 5, 5, 2, 1], 20, 20160, 403200 [14, 12, 9, 8, 5, 4, 4, 0], 14, 20160, 282240 [14, 12, 9, 8, 5, 4, 3, 1], 36, 40320, 1451520 [14, 12, 9, 8, 5, 4, 2, 2], 50, 20160, 1008000 [14, 12, 9, 8, 5, 3, 3, 2], 62, 20160, 1249920 [14, 12, 9, 8, 4, 4, 4, 1], 48, 6720, 322560 [14, 12, 9, 8, 4, 4, 3, 2], 82, 20160, 1653120 [14, 12, 9, 8, 4, 3, 3, 3], 102, 6720, 685440 [14, 12, 9, 7, 7, 5, 2, 0], 5, 20160, 100800 [14, 12, 9, 7, 7, 5, 1, 1], 5, 10080, 50400 [14, 12, 9, 7, 7, 4, 3, 0], 10, 20160, 201600 [14, 12, 9, 7, 7, 4, 2, 1], 20, 20160, 403200 [14, 12, 9, 7, 7, 3, 3, 1], 25, 10080, 252000 [14, 12, 9, 7, 7, 3, 2, 2], 35, 10080, 352800 [14, 12, 9, 7, 6, 6, 2, 0], 7, 20160, 141120 [14, 12, 9, 7, 6, 6, 1, 1], 7, 10080, 70560 [14, 12, 9, 7, 6, 5, 3, 0], 18, 40320, 725760 [14, 12, 9, 7, 6, 5, 2, 1], 36, 40320, 1451520 [14, 12, 9, 7, 6, 4, 4, 0], 25, 20160, 504000 [14, 12, 9, 7, 6, 4, 3, 1], 65, 40320, 2620800 [14, 12, 9, 7, 6, 4, 2, 2], 90, 20160, 1814400 [14, 12, 9, 7, 6, 3, 3, 2], 112, 20160, 2257920 [14, 12, 9, 7, 5, 5, 4, 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8, 8, 8, 8, 7, 7, 2], 16457832, 168, 2764915776 [8, 8, 8, 8, 8, 7, 6, 3], 35091153, 336, 11790627408 [8, 8, 8, 8, 8, 7, 5, 4], 50328549, 336, 16910392464 [8, 8, 8, 8, 8, 6, 6, 4], 59218704, 168, 9948742272 [8, 8, 8, 8, 8, 6, 5, 5], 70098840, 168, 11776605120 [8, 8, 8, 8, 7, 7, 7, 3], 41162319, 280, 11525449320 [8, 8, 8, 8, 7, 7, 6, 4], 69315975, 840, 58225419000 [8, 8, 8, 8, 7, 7, 5, 5], 82001736, 420, 34440729120 [8, 8, 8, 8, 7, 6, 6, 5], 96232869, 840, 80835609960 [8, 8, 8, 8, 6, 6, 6, 6], 112837032, 70, 7898592240 [8, 8, 8, 7, 7, 7, 7, 4], 81079146, 280, 22702160880 [8, 8, 8, 7, 7, 7, 6, 5], 112418424, 1120, 125908634880 [8, 8, 8, 7, 7, 6, 6, 6], 131727069, 560, 73767158640 [8, 8, 7, 7, 7, 7, 7, 5], 131244039, 168, 22048998552 [8, 8, 7, 7, 7, 7, 6, 6], 153686790, 420, 64548451800 [8, 7, 7, 7, 7, 7, 7, 6], 179202573, 56, 10035344088 [7, 7, 7, 7, 7, 7, 7, 7], 208836207, 1, 208836207 Sum of numbers in last column=, 22876792454961 The most probable vector is, [10, 9, 8, 7, 7, 6, 5, 4], which arises in, 445481588160, out of, 3, to the power, 28 that equals, 22876792454961, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 8, players The sole champion score-vector is, [10, 9, 8, 7, 7, 6, 5, 4] the corresponding record is, 445481588160 ----------------------------------------------------------------------------\ ----- 9, Players. Symmetric Function:, 9 / 9 \ --------' | --------' | ' | | |' | | 2 2 | | | | | | (a[i] + a[i] a[j] + a[j] )| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [16, 14, 12, 10, 8, 6, 4, 2, 0], 1, 362880, 362880 [16, 14, 12, 10, 8, 6, 4, 1, 1], 1, 181440, 181440 [16, 14, 12, 10, 8, 6, 3, 3, 0], 1, 181440, 181440 [16, 14, 12, 10, 8, 6, 3, 2, 1], 2, 362880, 725760 [16, 14, 12, 10, 8, 6, 2, 2, 2], 3, 60480, 181440 [16, 14, 12, 10, 8, 5, 5, 2, 0], 1, 181440, 181440 [16, 14, 12, 10, 8, 5, 5, 1, 1], 1, 90720, 90720 [16, 14, 12, 10, 8, 5, 4, 3, 0], 2, 362880, 725760 [16, 14, 12, 10, 8, 5, 4, 2, 1], 4, 362880, 1451520 [16, 14, 12, 10, 8, 5, 3, 3, 1], 5, 181440, 907200 [16, 14, 12, 10, 8, 5, 3, 2, 2], 7, 181440, 1270080 [16, 14, 12, 10, 8, 4, 4, 4, 0], 3, 60480, 181440 [16, 14, 12, 10, 8, 4, 4, 3, 1], 7, 181440, 1270080 [16, 14, 12, 10, 8, 4, 4, 2, 2], 10, 90720, 907200 [16, 14, 12, 10, 8, 4, 3, 3, 2], 12, 181440, 2177280 [16, 14, 12, 10, 8, 3, 3, 3, 3], 15, 15120, 226800 [16, 14, 12, 10, 7, 7, 4, 2, 0], 1, 181440, 181440 [16, 14, 12, 10, 7, 7, 4, 1, 1], 1, 90720, 90720 [16, 14, 12, 10, 7, 7, 3, 3, 0], 1, 90720, 90720 [16, 14, 12, 10, 7, 7, 3, 2, 1], 2, 181440, 362880 [16, 14, 12, 10, 7, 7, 2, 2, 2], 3, 30240, 90720 [16, 14, 12, 10, 7, 6, 5, 2, 0], 2, 362880, 725760 [16, 14, 12, 10, 7, 6, 5, 1, 1], 2, 181440, 362880 [16, 14, 12, 10, 7, 6, 4, 3, 0], 4, 362880, 1451520 [16, 14, 12, 10, 7, 6, 4, 2, 1], 8, 362880, 2903040 [16, 14, 12, 10, 7, 6, 3, 3, 1], 10, 181440, 1814400 [16, 14, 12, 10, 7, 6, 3, 2, 2], 14, 181440, 2540160 [16, 14, 12, 10, 7, 5, 5, 3, 0], 5, 181440, 907200 [16, 14, 12, 10, 7, 5, 5, 2, 1], 10, 181440, 1814400 [16, 14, 12, 10, 7, 5, 4, 4, 0], 7, 181440, 1270080 [16, 14, 12, 10, 7, 5, 4, 3, 1], 18, 362880, 6531840 [16, 14, 12, 10, 7, 5, 4, 2, 2], 25, 181440, 4536000 [16, 14, 12, 10, 7, 5, 3, 3, 2], 31, 181440, 5624640 [16, 14, 12, 10, 7, 4, 4, 4, 1], 24, 60480, 1451520 [16, 14, 12, 10, 7, 4, 4, 3, 2], 41, 181440, 7439040 [16, 14, 12, 10, 7, 4, 3, 3, 3], 51, 60480, 3084480 [16, 14, 12, 10, 6, 6, 6, 2, 0], 3, 60480, 181440 [16, 14, 12, 10, 6, 6, 6, 1, 1], 3, 30240, 90720 [16, 14, 12, 10, 6, 6, 5, 3, 0], 7, 181440, 1270080 [16, 14, 12, 10, 6, 6, 5, 2, 1], 14, 181440, 2540160 [16, 14, 12, 10, 6, 6, 4, 4, 0], 10, 90720, 907200 [16, 14, 12, 10, 6, 6, 4, 3, 1], 25, 181440, 4536000 [16, 14, 12, 10, 6, 6, 4, 2, 2], 35, 90720, 3175200 [16, 14, 12, 10, 6, 6, 3, 3, 2], 43, 90720, 3900960 [16, 14, 12, 10, 6, 5, 5, 4, 0], 12, 181440, 2177280 [16, 14, 12, 10, 6, 5, 5, 3, 1], 31, 181440, 5624640 [16, 14, 12, 10, 6, 5, 5, 2, 2], 43, 90720, 3900960 [16, 14, 12, 10, 6, 5, 4, 4, 1], 41, 181440, 7439040 [16, 14, 12, 10, 6, 5, 4, 3, 2], 70, 362880, 25401600 [16, 14, 12, 10, 6, 5, 3, 3, 3], 87, 60480, 5261760 [16, 14, 12, 10, 6, 4, 4, 4, 2], 90, 60480, 5443200 [16, 14, 12, 10, 6, 4, 4, 3, 3], 111, 90720, 10069920 [16, 14, 12, 10, 5, 5, 5, 5, 0], 15, 15120, 226800 [16, 14, 12, 10, 5, 5, 5, 4, 1], 51, 60480, 3084480 [16, 14, 12, 10, 5, 5, 5, 3, 2], 87, 60480, 5261760 [16, 14, 12, 10, 5, 5, 4, 4, 2], 111, 90720, 10069920 [16, 14, 12, 10, 5, 5, 4, 3, 3], 138, 90720, 12519360 [16, 14, 12, 10, 5, 4, 4, 4, 3], 174, 60480, 10523520 [16, 14, 12, 10, 4, 4, 4, 4, 4], 219, 3024, 662256 [16, 14, 12, 9, 9, 6, 4, 2, 0], 1, 181440, 181440 [16, 14, 12, 9, 9, 6, 4, 1, 1], 1, 90720, 90720 [16, 14, 12, 9, 9, 6, 3, 3, 0], 1, 90720, 90720 [16, 14, 12, 9, 9, 6, 3, 2, 1], 2, 181440, 362880 [16, 14, 12, 9, 9, 6, 2, 2, 2], 3, 30240, 90720 [16, 14, 12, 9, 9, 5, 5, 2, 0], 1, 90720, 90720 [16, 14, 12, 9, 9, 5, 5, 1, 1], 1, 45360, 45360 [16, 14, 12, 9, 9, 5, 4, 3, 0], 2, 181440, 362880 [16, 14, 12, 9, 9, 5, 4, 2, 1], 4, 181440, 725760 [16, 14, 12, 9, 9, 5, 3, 3, 1], 5, 90720, 453600 [16, 14, 12, 9, 9, 5, 3, 2, 2], 7, 90720, 635040 [16, 14, 12, 9, 9, 4, 4, 4, 0], 3, 30240, 90720 [16, 14, 12, 9, 9, 4, 4, 3, 1], 7, 90720, 635040 [16, 14, 12, 9, 9, 4, 4, 2, 2], 10, 45360, 453600 [16, 14, 12, 9, 9, 4, 3, 3, 2], 12, 90720, 1088640 [16, 14, 12, 9, 9, 3, 3, 3, 3], 15, 7560, 113400 [16, 14, 12, 9, 8, 7, 4, 2, 0], 2, 362880, 725760 [16, 14, 12, 9, 8, 7, 4, 1, 1], 2, 181440, 362880 [16, 14, 12, 9, 8, 7, 3, 3, 0], 2, 181440, 362880 [16, 14, 12, 9, 8, 7, 3, 2, 1], 4, 362880, 1451520 [16, 14, 12, 9, 8, 7, 2, 2, 2], 6, 60480, 362880 [16, 14, 12, 9, 8, 6, 5, 2, 0], 4, 362880, 1451520 [16, 14, 12, 9, 8, 6, 5, 1, 1], 4, 181440, 725760 [16, 14, 12, 9, 8, 6, 4, 3, 0], 8, 362880, 2903040 [16, 14, 12, 9, 8, 6, 4, 2, 1], 16, 362880, 5806080 [16, 14, 12, 9, 8, 6, 3, 3, 1], 20, 181440, 3628800 [16, 14, 12, 9, 8, 6, 3, 2, 2], 28, 181440, 5080320 [16, 14, 12, 9, 8, 5, 5, 3, 0], 10, 181440, 1814400 [16, 14, 12, 9, 8, 5, 5, 2, 1], 20, 181440, 3628800 [16, 14, 12, 9, 8, 5, 4, 4, 0], 14, 181440, 2540160 [16, 14, 12, 9, 8, 5, 4, 3, 1], 36, 362880, 13063680 [16, 14, 12, 9, 8, 5, 4, 2, 2], 50, 181440, 9072000 [16, 14, 12, 9, 8, 5, 3, 3, 2], 62, 181440, 11249280 [16, 14, 12, 9, 8, 4, 4, 4, 1], 48, 60480, 2903040 [16, 14, 12, 9, 8, 4, 4, 3, 2], 82, 181440, 14878080 [16, 14, 12, 9, 8, 4, 3, 3, 3], 102, 60480, 6168960 [16, 14, 12, 9, 7, 7, 5, 2, 0], 5, 181440, 907200 [16, 14, 12, 9, 7, 7, 5, 1, 1], 5, 90720, 453600 [16, 14, 12, 9, 7, 7, 4, 3, 0], 10, 181440, 1814400 [16, 14, 12, 9, 7, 7, 4, 2, 1], 20, 181440, 3628800 [16, 14, 12, 9, 7, 7, 3, 3, 1], 25, 90720, 2268000 [16, 14, 12, 9, 7, 7, 3, 2, 2], 35, 90720, 3175200 [16, 14, 12, 9, 7, 6, 6, 2, 0], 7, 181440, 1270080 [16, 14, 12, 9, 7, 6, 6, 1, 1], 7, 90720, 635040 [16, 14, 12, 9, 7, 6, 5, 3, 0], 18, 362880, 6531840 [16, 14, 12, 9, 7, 6, 5, 2, 1], 36, 362880, 13063680 [16, 14, 12, 9, 7, 6, 4, 4, 0], 25, 181440, 4536000 [16, 14, 12, 9, 7, 6, 4, 3, 1], 65, 362880, 23587200 [16, 14, 12, 9, 7, 6, 4, 2, 2], 90, 181440, 16329600 [16, 14, 12, 9, 7, 6, 3, 3, 2], 112, 181440, 20321280 [16, 14, 12, 9, 7, 5, 5, 4, 0], 31, 181440, 5624640 [16, 14, 12, 9, 7, 5, 5, 3, 1], 82, 181440, 14878080 [16, 14, 12, 9, 7, 5, 5, 2, 2], 113, 90720, 10251360 [16, 14, 12, 9, 7, 5, 4, 4, 1], 108, 181440, 19595520 [16, 14, 12, 9, 7, 5, 4, 3, 2], 185, 362880, 67132800 [16, 14, 12, 9, 7, 5, 3, 3, 3], 231, 60480, 13970880 [16, 14, 12, 9, 7, 4, 4, 4, 2], 237, 60480, 14333760 [16, 14, 12, 9, 7, 4, 4, 3, 3], 295, 90720, 26762400 [16, 14, 12, 9, 6, 6, 6, 3, 0], 24, 60480, 1451520 [16, 14, 12, 9, 6, 6, 6, 2, 1], 48, 60480, 2903040 [16, 14, 12, 9, 6, 6, 5, 4, 0], 41, 181440, 7439040 [16, 14, 12, 9, 6, 6, 5, 3, 1], 108, 181440, 19595520 [16, 14, 12, 9, 6, 6, 5, 2, 2], 149, 90720, 13517280 [16, 14, 12, 9, 6, 6, 4, 4, 1], 142, 90720, 12882240 [16, 14, 12, 9, 6, 6, 4, 3, 2], 243, 181440, 44089920 [16, 14, 12, 9, 6, 6, 3, 3, 3], 303, 30240, 9162720 [16, 14, 12, 9, 6, 5, 5, 5, 0], 51, 60480, 3084480 [16, 14, 12, 9, 6, 5, 5, 4, 1], 176, 181440, 31933440 [16, 14, 12, 9, 6, 5, 5, 3, 2], 301, 181440, 54613440 [16, 14, 12, 9, 6, 5, 4, 4, 2], 382, 181440, 69310080 [16, 14, 12, 9, 6, 5, 4, 3, 3], 476, 181440, 86365440 [16, 14, 12, 9, 6, 4, 4, 4, 3], 597, 60480, 36106560 [16, 14, 12, 9, 5, 5, 5, 5, 1], 219, 15120, 3311280 [16, 14, 12, 9, 5, 5, 5, 4, 2], 471, 60480, 28486080 [16, 14, 12, 9, 5, 5, 5, 3, 3], 588, 30240, 17781120 [16, 14, 12, 9, 5, 5, 4, 4, 3], 735, 90720, 66679200 [16, 14, 12, 9, 5, 4, 4, 4, 4], 915, 15120, 13834800 [16, 14, 12, 8, 8, 8, 4, 2, 0], 3, 60480, 181440 [16, 14, 12, 8, 8, 8, 4, 1, 1], 3, 30240, 90720 [16, 14, 12, 8, 8, 8, 3, 3, 0], 3, 30240, 90720 [16, 14, 12, 8, 8, 8, 3, 2, 1], 6, 60480, 362880 [16, 14, 12, 8, 8, 8, 2, 2, 2], 9, 10080, 90720 [16, 14, 12, 8, 8, 7, 5, 2, 0], 7, 181440, 1270080 [16, 14, 12, 8, 8, 7, 5, 1, 1], 7, 90720, 635040 [16, 14, 12, 8, 8, 7, 4, 3, 0], 14, 181440, 2540160 [16, 14, 12, 8, 8, 7, 4, 2, 1], 28, 181440, 5080320 [16, 14, 12, 8, 8, 7, 3, 3, 1], 35, 90720, 3175200 [16, 14, 12, 8, 8, 7, 3, 2, 2], 49, 90720, 4445280 [16, 14, 12, 8, 8, 6, 6, 2, 0], 10, 90720, 907200 [16, 14, 12, 8, 8, 6, 6, 1, 1], 10, 45360, 453600 [16, 14, 12, 8, 8, 6, 5, 3, 0], 25, 181440, 4536000 [16, 14, 12, 8, 8, 6, 5, 2, 1], 50, 181440, 9072000 [16, 14, 12, 8, 8, 6, 4, 4, 0], 35, 90720, 3175200 [16, 14, 12, 8, 8, 6, 4, 3, 1], 90, 181440, 16329600 [16, 14, 12, 8, 8, 6, 4, 2, 2], 125, 90720, 11340000 [16, 14, 12, 8, 8, 6, 3, 3, 2], 155, 90720, 14061600 [16, 14, 12, 8, 8, 5, 5, 4, 0], 43, 90720, 3900960 [16, 14, 12, 8, 8, 5, 5, 3, 1], 113, 90720, 10251360 [16, 14, 12, 8, 8, 5, 5, 2, 2], 156, 45360, 7076160 [16, 14, 12, 8, 8, 5, 4, 4, 1], 149, 90720, 13517280 [16, 14, 12, 8, 8, 5, 4, 3, 2], 255, 181440, 46267200 [16, 14, 12, 8, 8, 5, 3, 3, 3], 318, 30240, 9616320 [16, 14, 12, 8, 8, 4, 4, 4, 2], 327, 30240, 9888480 [16, 14, 12, 8, 8, 4, 4, 3, 3], 406, 45360, 18416160 [16, 14, 12, 8, 7, 7, 6, 2, 0], 12, 181440, 2177280 [16, 14, 12, 8, 7, 7, 6, 1, 1], 12, 90720, 1088640 [16, 14, 12, 8, 7, 7, 5, 3, 0], 31, 181440, 5624640 [16, 14, 12, 8, 7, 7, 5, 2, 1], 62, 181440, 11249280 [16, 14, 12, 8, 7, 7, 4, 4, 0], 43, 90720, 3900960 [16, 14, 12, 8, 7, 7, 4, 3, 1], 112, 181440, 20321280 [16, 14, 12, 8, 7, 7, 4, 2, 2], 155, 90720, 14061600 [16, 14, 12, 8, 7, 7, 3, 3, 2], 193, 90720, 17508960 [16, 14, 12, 8, 7, 6, 6, 3, 0], 41, 181440, 7439040 [16, 14, 12, 8, 7, 6, 6, 2, 1], 82, 181440, 14878080 [16, 14, 12, 8, 7, 6, 5, 4, 0], 70, 362880, 25401600 [16, 14, 12, 8, 7, 6, 5, 3, 1], 185, 362880, 67132800 [16, 14, 12, 8, 7, 6, 5, 2, 2], 255, 181440, 46267200 [16, 14, 12, 8, 7, 6, 4, 4, 1], 243, 181440, 44089920 [16, 14, 12, 8, 7, 6, 4, 3, 2], 416, 362880, 150958080 [16, 14, 12, 8, 7, 6, 3, 3, 3], 519, 60480, 31389120 [16, 14, 12, 8, 7, 5, 5, 5, 0], 87, 60480, 5261760 [16, 14, 12, 8, 7, 5, 5, 4, 1], 301, 181440, 54613440 [16, 14, 12, 8, 7, 5, 5, 3, 2], 515, 181440, 93441600 [16, 14, 12, 8, 7, 5, 4, 4, 2], 653, 181440, 118480320 [16, 14, 12, 8, 7, 5, 4, 3, 3], 814, 181440, 147692160 [16, 14, 12, 8, 7, 4, 4, 4, 3], 1020, 60480, 61689600 [16, 14, 12, 8, 6, 6, 6, 4, 0], 90, 60480, 5443200 [16, 14, 12, 8, 6, 6, 6, 3, 1], 237, 60480, 14333760 [16, 14, 12, 8, 6, 6, 6, 2, 2], 327, 30240, 9888480 [16, 14, 12, 8, 6, 6, 5, 5, 0], 111, 90720, 10069920 [16, 14, 12, 8, 6, 6, 5, 4, 1], 382, 181440, 69310080 [16, 14, 12, 8, 6, 6, 5, 3, 2], 653, 181440, 118480320 [16, 14, 12, 8, 6, 6, 4, 4, 2], 826, 90720, 74934720 [16, 14, 12, 8, 6, 6, 4, 3, 3], 1028, 90720, 93260160 [16, 14, 12, 8, 6, 5, 5, 5, 1], 471, 60480, 28486080 [16, 14, 12, 8, 6, 5, 5, 4, 2], 1009, 181440, 183072960 [16, 14, 12, 8, 6, 5, 5, 3, 3], 1258, 90720, 114125760 [16, 14, 12, 8, 6, 5, 4, 4, 3], 1566, 181440, 284135040 [16, 14, 12, 8, 6, 4, 4, 4, 4], 1941, 15120, 29347920 [16, 14, 12, 8, 5, 5, 5, 5, 2], 1233, 15120, 18642960 [16, 14, 12, 8, 5, 5, 5, 4, 3], 1908, 60480, 115395840 [16, 14, 12, 8, 5, 5, 4, 4, 4], 2358, 30240, 71305920 [16, 14, 12, 7, 7, 7, 7, 2, 0], 15, 15120, 226800 [16, 14, 12, 7, 7, 7, 7, 1, 1], 15, 7560, 113400 [16, 14, 12, 7, 7, 7, 6, 3, 0], 51, 60480, 3084480 [16, 14, 12, 7, 7, 7, 6, 2, 1], 102, 60480, 6168960 [16, 14, 12, 7, 7, 7, 5, 4, 0], 87, 60480, 5261760 [16, 14, 12, 7, 7, 7, 5, 3, 1], 231, 60480, 13970880 [16, 14, 12, 7, 7, 7, 5, 2, 2], 318, 30240, 9616320 [16, 14, 12, 7, 7, 7, 4, 4, 1], 303, 30240, 9162720 [16, 14, 12, 7, 7, 7, 4, 3, 2], 519, 60480, 31389120 [16, 14, 12, 7, 7, 7, 3, 3, 3], 648, 10080, 6531840 [16, 14, 12, 7, 7, 6, 6, 4, 0], 111, 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7, 6, 5], 24204459681, 15120, 365971430376720 [10, 10, 9, 9, 9, 7, 6, 6, 6], 28206571905, 5040, 142161122401200 [10, 10, 9, 9, 8, 8, 8, 8, 2], 4313060892, 3780, 16303370171760 [10, 10, 9, 9, 8, 8, 8, 7, 3], 10560849858, 15120, 159680049852960 [10, 10, 9, 9, 8, 8, 8, 6, 4], 17571181791, 15120, 265676268679920 [10, 10, 9, 9, 8, 8, 8, 5, 5], 20718715359, 7560, 156633488114040 [10, 10, 9, 9, 8, 8, 7, 7, 4], 20375407560, 22680, 462114243460800 [10, 10, 9, 9, 8, 8, 7, 6, 5], 27988130220, 45360, 1269541586779200 [10, 10, 9, 9, 8, 8, 6, 6, 6], 32599994544, 7560, 246455958752640 [10, 10, 9, 9, 8, 7, 7, 7, 5], 32396179509, 15120, 489830234176080 [10, 10, 9, 9, 8, 7, 7, 6, 6], 37712068050, 22680, 855309703374000 [10, 10, 9, 9, 7, 7, 7, 7, 6], 43602783678, 3780, 164818522302840 [10, 10, 9, 8, 8, 8, 8, 8, 3], 12239614791, 1512, 18506297563992 [10, 10, 9, 8, 8, 8, 8, 7, 4], 23568705681, 7560, 178179414948360 [10, 10, 9, 8, 8, 8, 8, 6, 5], 32345390640, 7560, 244531153238400 [10, 10, 9, 8, 8, 8, 7, 7, 5], 37420913040, 15120, 565804205164800 [10, 10, 9, 8, 8, 8, 7, 6, 6], 43541235444, 15120, 658343479913280 [10, 10, 9, 8, 8, 7, 7, 7, 6], 50318617374, 15120, 760817494694880 [10, 10, 9, 8, 7, 7, 7, 7, 7], 58121991216, 1512, 87880450718592 [10, 10, 8, 8, 8, 8, 8, 8, 4], 27247498056, 252, 6866369510112 [10, 10, 8, 8, 8, 8, 8, 7, 5], 43202630712, 1512, 65322377636544 [10, 10, 8, 8, 8, 8, 8, 6, 6], 50245944024, 756, 37985933682144 [10, 10, 8, 8, 8, 8, 7, 7, 6], 58040207820, 3780, 219391985559600 [10, 10, 8, 8, 8, 7, 7, 7, 7], 67011006582, 1260, 84433868293320 [10, 9, 9, 9, 9, 9, 9, 8, 0], 179202573, 504, 90318096792 [10, 9, 9, 9, 9, 9, 9, 7, 1], 1078683381, 504, 543656424024 [10, 9, 9, 9, 9, 9, 9, 6, 2], 3173573880, 504, 1599481235520 [10, 9, 9, 9, 9, 9, 9, 5, 3], 5721168546, 504, 2883468947184 [10, 9, 9, 9, 9, 9, 9, 4, 4], 6916578237, 252, 1742977715724 [10, 9, 9, 9, 9, 9, 8, 8, 1], 1258083135, 1512, 1902221700120 [10, 9, 9, 9, 9, 9, 8, 7, 2], 4309685748, 3024, 13032489701952 [10, 9, 9, 9, 9, 9, 8, 6, 3], 9083024199, 3024, 27467065177776 [10, 9, 9, 9, 9, 9, 8, 5, 4], 12956684661, 3024, 39181014414864 [10, 9, 9, 9, 9, 9, 7, 7, 3], 10562621553, 1512, 15970683788136 [10, 9, 9, 9, 9, 9, 7, 6, 4], 17583522705, 3024, 53172572659920 [10, 9, 9, 9, 9, 9, 7, 5, 5], 20736700182, 1512, 31353890675184 [10, 9, 9, 9, 9, 9, 6, 6, 5], 24190980444, 1512, 36576762431328 [10, 9, 9, 9, 9, 8, 8, 8, 2], 5007430638, 2520, 12618725207760 [10, 9, 9, 9, 9, 8, 8, 7, 3], 12242537748, 7560, 92553585374880 [10, 9, 9, 9, 9, 8, 8, 6, 4], 20351504448, 7560, 153857373626880 [10, 9, 9, 9, 9, 8, 8, 5, 5], 23990950845, 3780, 90685794194100 [10, 9, 9, 9, 9, 8, 7, 7, 4], 23589504234, 7560, 178336652009040 [10, 9, 9, 9, 9, 8, 7, 6, 5], 32383649934, 15120, 489640787002080 [10, 9, 9, 9, 9, 8, 6, 6, 6], 37707026958, 2520, 95021707934160 [10, 9, 9, 9, 9, 7, 7, 7, 5], 37468958562, 2520, 94421775576240 [10, 9, 9, 9, 9, 7, 7, 6, 6], 43602783678, 3780, 164818522302840 [10, 9, 9, 9, 8, 8, 8, 8, 3], 14181246324, 2520, 35736740736480 [10, 9, 9, 9, 8, 8, 8, 7, 4], 27272890467, 10080, 274910735907360 [10, 9, 9, 9, 8, 8, 8, 6, 5], 37406842398, 10080, 377060971371840 [10, 9, 9, 9, 8, 8, 7, 7, 5], 43259697657, 15120, 654086628573840 [10, 9, 9, 9, 8, 8, 7, 6, 6], 50318617374, 15120, 760817494694880 [10, 9, 9, 9, 8, 7, 7, 7, 6], 58129087302, 10080, 585941200004160 [10, 9, 9, 9, 7, 7, 7, 7, 7], 67118974632, 504, 33827963214528 [10, 9, 9, 8, 8, 8, 8, 8, 4], 31514578677, 1512, 47650042959624 [10, 9, 9, 8, 8, 8, 8, 7, 5], 49920235704, 7560, 377396981922240 [10, 9, 9, 8, 8, 8, 8, 6, 6], 58040207820, 3780, 219391985559600 [10, 9, 9, 8, 8, 8, 7, 7, 6], 67018958946, 15120, 1013326659263520 [10, 9, 9, 8, 8, 7, 7, 7, 7], 77349699891, 3780, 292381865587980 [10, 9, 8, 8, 8, 8, 8, 8, 5], 57578201001, 504, 29019413304504 [10, 9, 8, 8, 8, 8, 8, 7, 6], 77232431340, 3024, 233550872372160 [10, 9, 8, 8, 8, 8, 7, 7, 7], 89099404866, 2520, 224530500262320 [10, 8, 8, 8, 8, 8, 8, 8, 6], 88962207264, 72, 6405278923008 [10, 8, 8, 8, 8, 8, 8, 7, 7], 102588687603, 252, 25852349275956 [9, 9, 9, 9, 9, 9, 9, 9, 0], 208836207, 9, 1879525863 [9, 9, 9, 9, 9, 9, 9, 8, 1], 1463254218, 72, 105354303696 [9, 9, 9, 9, 9, 9, 9, 7, 2], 5003495055, 72, 360251643960 [9, 9, 9, 9, 9, 9, 9, 6, 3], 10533979008, 72, 758446488576 [9, 9, 9, 9, 9, 9, 9, 5, 4], 15018881157, 72, 1081359443304 [9, 9, 9, 9, 9, 9, 8, 8, 2], 5810438313, 252, 1464230454876 [9, 9, 9, 9, 9, 9, 8, 7, 3], 14184387603, 504, 7148931351912 [9, 9, 9, 9, 9, 9, 8, 6, 4], 23559150213, 504, 11873811707352 [9, 9, 9, 9, 9, 9, 8, 5, 5], 27765117981, 252, 6996809731212 [9, 9, 9, 9, 9, 9, 7, 7, 4], 27296212545, 252, 6878645561340 [9, 9, 9, 9, 9, 9, 7, 6, 5], 37449892620, 504, 18874745880480 [9, 9, 9, 9, 9, 9, 6, 6, 6], 43591493412, 84, 3661685446608 [9, 9, 9, 9, 9, 8, 8, 8, 3], 16422295545, 504, 8276836954680 [9, 9, 9, 9, 9, 8, 8, 7, 4], 31543053003, 1512, 47693096140536 [9, 9, 9, 9, 9, 8, 8, 6, 5], 43238294379, 1512, 65376301101048 [9, 9, 9, 9, 9, 8, 7, 7, 5], 49984425063, 1512, 75576450695256 [9, 9, 9, 9, 9, 8, 7, 6, 6], 58121991216, 1512, 87880450718592 [9, 9, 9, 9, 9, 7, 7, 7, 6], 67118974632, 504, 33827963214528 [9, 9, 9, 9, 8, 8, 8, 8, 4], 36431680059, 630, 22951958437170 [9, 9, 9, 9, 8, 8, 8, 7, 5], 57654160308, 2520, 145288483976160 [9, 9, 9, 9, 8, 8, 8, 6, 6], 67011006582, 1260, 84433868293320 [9, 9, 9, 9, 8, 8, 7, 7, 6], 77349699891, 3780, 292381865587980 [9, 9, 9, 9, 8, 7, 7, 7, 7], 89241385977, 630, 56222073165510 [9, 9, 9, 8, 8, 8, 8, 8, 5], 66469170537, 504, 33500461950648 [9, 9, 9, 8, 8, 8, 8, 7, 6], 89099404866, 2520, 224530500262320 [9, 9, 9, 8, 8, 8, 7, 7, 7], 102754297710, 1680, 172627220152800 [9, 9, 8, 8, 8, 8, 8, 8, 6], 102588687603, 252, 25852349275956 [9, 9, 8, 8, 8, 8, 8, 7, 7], 118262399985, 756, 89406374388660 [9, 8, 8, 8, 8, 8, 8, 8, 7], 136054026828, 72, 9795889931616 [8, 8, 8, 8, 8, 8, 8, 8, 8], 156458382975, 1, 156458382975 Sum of numbers in last column=, 150094635296999121 The most probable vector is, [11, 10, 9, 9, 8, 7, 7, 6, 5], which arises in, 1625794339095360, out of, 3, to the power, 36 that equals, 150094635296999121, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 9, players The sole champion score-vector is, [11, 10, 9, 9, 8, 7, 7, 6, 5] the corresponding record is, 1625794339095360 ------------------------------------------ In conclusion, the n=2 thorough n=, 9, of the champions are [12, 144, 8400, 1059480, 343900620, 445481588160, 1625794339095360] ------------------------------------------ This ends this article, that took, 218.034, seconds to produce.