All the Possible Score Vectors in a Round-Robin Tournament, and the Number o\ f Times each Shows Up for tournaments from Three to , 12, players. By Shalosh B. Ekhad In his beautiful article, "An American Tournament Treated by the Calculus of\ Symmetric Functions" Quarterly J. of Pure and Applied Mathematics. vol. XLIX, No. 193, 1920 The Combinatorial Giant, (and amazing human calculator), Major Percy Alexand\ er MacMahon considered round-robin tournaments for n players for n from 2 to 9, and for \ each found all the possible score vectors and, for each, the number of times it shows up In this article, we reproduce the tables done by the Major for n=3 to n=8 and go all the way to n=, 12 ----------------------------------------------------------------------------\ ----- 3 / 3 \ --------' | --------' | ' | | |' | | | 3, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [2, 1, 0], 1, 6, 6 [1, 1, 1], 2, 1, 2 Sum of numbers in last column=, 8 The most probable vector is, [2, 1, 0], which arises in, 6, out of 2 to the power, 3, that equals, 8, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 3, players The sole champion score-vector is, [2, 1, 0] the corresponding record is, 6 ----------------------------------------------------------------------------\ ----- 4 / 4 \ --------' | --------' | ' | | |' | | | 4, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [3, 2, 1, 0], 1, 24, 24 [3, 1, 1, 1], 2, 4, 8 [2, 2, 2, 0], 2, 4, 8 [2, 2, 1, 1], 4, 6, 24 Sum of numbers in last column=, 64 The most probable vector is, [2, 2, 1, 1], which arises in, 24, out of 2 to the power, 6, that equals, 64, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 4, players The set of champion score vectors is, {[2, 2, 1, 1], [3, 2, 1, 0]} the corresponding record is, 24 ----------------------------------------------------------------------------\ ----- 5 / 5 \ --------' | --------' | ' | | |' | | | 5, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [4, 3, 2, 1, 0], 1, 120, 120 [4, 3, 1, 1, 1], 2, 20, 40 [4, 2, 2, 2, 0], 2, 20, 40 [4, 2, 2, 1, 1], 4, 30, 120 [3, 3, 3, 1, 0], 2, 20, 40 [3, 3, 2, 2, 0], 4, 30, 120 [3, 3, 2, 1, 1], 8, 30, 240 [3, 2, 2, 2, 1], 14, 20, 280 [2, 2, 2, 2, 2], 24, 1, 24 Sum of numbers in last column=, 1024 The most probable vector is, [3, 2, 2, 2, 1], which arises in, 280, out of 2 to the power, 10, that equals, 1024, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 5, players The sole champion score-vector is, [3, 2, 2, 2, 1] the corresponding record is, 280 ----------------------------------------------------------------------------\ ----- 6 / 6 \ --------' | --------' | ' | | |' | | | 6, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [5, 4, 3, 2, 1, 0], 1, 720, 720 [5, 4, 3, 1, 1, 1], 2, 120, 240 [5, 4, 2, 2, 2, 0], 2, 120, 240 [5, 4, 2, 2, 1, 1], 4, 180, 720 [5, 3, 3, 3, 1, 0], 2, 120, 240 [5, 3, 3, 2, 2, 0], 4, 180, 720 [5, 3, 3, 2, 1, 1], 8, 180, 1440 [5, 3, 2, 2, 2, 1], 14, 120, 1680 [5, 2, 2, 2, 2, 2], 24, 6, 144 [4, 4, 4, 2, 1, 0], 2, 120, 240 [4, 4, 4, 1, 1, 1], 4, 20, 80 [4, 4, 3, 3, 1, 0], 4, 180, 720 [4, 4, 3, 2, 2, 0], 8, 180, 1440 [4, 4, 3, 2, 1, 1], 16, 180, 2880 [4, 4, 2, 2, 2, 1], 28, 60, 1680 [4, 3, 3, 3, 2, 0], 14, 120, 1680 [4, 3, 3, 3, 1, 1], 28, 60, 1680 [4, 3, 3, 2, 2, 1], 48, 180, 8640 [4, 3, 2, 2, 2, 2], 80, 30, 2400 [3, 3, 3, 3, 3, 0], 24, 6, 144 [3, 3, 3, 3, 2, 1], 80, 30, 2400 [3, 3, 3, 2, 2, 2], 132, 20, 2640 Sum of numbers in last column=, 32768 The most probable vector is, [4, 3, 3, 2, 2, 1], which arises in, 8640, out of 2 to the power, 15, that equals, 32768, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 6, players The sole champion score-vector is, [4, 3, 3, 2, 2, 1] the corresponding record is, 8640 ----------------------------------------------------------------------------\ ----- 7 / 7 \ --------' | --------' | ' | | |' | | | 7, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [6, 5, 4, 3, 2, 1, 0], 1, 5040, 5040 [6, 5, 4, 3, 1, 1, 1], 2, 840, 1680 [6, 5, 4, 2, 2, 2, 0], 2, 840, 1680 [6, 5, 4, 2, 2, 1, 1], 4, 1260, 5040 [6, 5, 3, 3, 3, 1, 0], 2, 840, 1680 [6, 5, 3, 3, 2, 2, 0], 4, 1260, 5040 [6, 5, 3, 3, 2, 1, 1], 8, 1260, 10080 [6, 5, 3, 2, 2, 2, 1], 14, 840, 11760 [6, 5, 2, 2, 2, 2, 2], 24, 42, 1008 [6, 4, 4, 4, 2, 1, 0], 2, 840, 1680 [6, 4, 4, 4, 1, 1, 1], 4, 140, 560 [6, 4, 4, 3, 3, 1, 0], 4, 1260, 5040 [6, 4, 4, 3, 2, 2, 0], 8, 1260, 10080 [6, 4, 4, 3, 2, 1, 1], 16, 1260, 20160 [6, 4, 4, 2, 2, 2, 1], 28, 420, 11760 [6, 4, 3, 3, 3, 2, 0], 14, 840, 11760 [6, 4, 3, 3, 3, 1, 1], 28, 420, 11760 [6, 4, 3, 3, 2, 2, 1], 48, 1260, 60480 [6, 4, 3, 2, 2, 2, 2], 80, 210, 16800 [6, 3, 3, 3, 3, 3, 0], 24, 42, 1008 [6, 3, 3, 3, 3, 2, 1], 80, 210, 16800 [6, 3, 3, 3, 2, 2, 2], 132, 140, 18480 [5, 5, 5, 3, 2, 1, 0], 2, 840, 1680 [5, 5, 5, 3, 1, 1, 1], 4, 140, 560 [5, 5, 5, 2, 2, 2, 0], 4, 140, 560 [5, 5, 5, 2, 2, 1, 1], 8, 210, 1680 [5, 5, 4, 4, 2, 1, 0], 4, 1260, 5040 [5, 5, 4, 4, 1, 1, 1], 8, 210, 1680 [5, 5, 4, 3, 3, 1, 0], 8, 1260, 10080 [5, 5, 4, 3, 2, 2, 0], 16, 1260, 20160 [5, 5, 4, 3, 2, 1, 1], 32, 1260, 40320 [5, 5, 4, 2, 2, 2, 1], 56, 420, 23520 [5, 5, 3, 3, 3, 2, 0], 28, 420, 11760 [5, 5, 3, 3, 3, 1, 1], 56, 210, 11760 [5, 5, 3, 3, 2, 2, 1], 96, 630, 60480 [5, 5, 3, 2, 2, 2, 2], 160, 105, 16800 [5, 4, 4, 4, 3, 1, 0], 14, 840, 11760 [5, 4, 4, 4, 2, 2, 0], 28, 420, 11760 [5, 4, 4, 4, 2, 1, 1], 56, 420, 23520 [5, 4, 4, 3, 3, 2, 0], 48, 1260, 60480 [5, 4, 4, 3, 3, 1, 1], 96, 630, 60480 [5, 4, 4, 3, 2, 2, 1], 164, 1260, 206640 [5, 4, 4, 2, 2, 2, 2], 272, 105, 28560 [5, 4, 3, 3, 3, 3, 0], 80, 210, 16800 [5, 4, 3, 3, 3, 2, 1], 266, 840, 223440 [5, 4, 3, 3, 2, 2, 2], 436, 420, 183120 [5, 3, 3, 3, 3, 3, 1], 424, 42, 17808 [5, 3, 3, 3, 3, 2, 2], 688, 105, 72240 [4, 4, 4, 4, 4, 1, 0], 24, 42, 1008 [4, 4, 4, 4, 3, 2, 0], 80, 210, 16800 [4, 4, 4, 4, 3, 1, 1], 160, 105, 16800 [4, 4, 4, 4, 2, 2, 1], 272, 105, 28560 [4, 4, 4, 3, 3, 3, 0], 132, 140, 18480 [4, 4, 4, 3, 3, 2, 1], 436, 420, 183120 [4, 4, 4, 3, 2, 2, 2], 712, 140, 99680 [4, 4, 3, 3, 3, 3, 1], 688, 105, 72240 [4, 4, 3, 3, 3, 2, 2], 1112, 210, 233520 [4, 3, 3, 3, 3, 3, 2], 1720, 42, 72240 [3, 3, 3, 3, 3, 3, 3], 2640, 1, 2640 Sum of numbers in last column=, 2097152 The most probable vector is, [4, 4, 3, 3, 3, 2, 2], which arises in, 233520, out of 2 to the power, 21, that equals, 2097152, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 7, players The sole champion score-vector is, [4, 4, 3, 3, 3, 2, 2] the corresponding record is, 233520 ----------------------------------------------------------------------------\ ----- 8 / 8 \ --------' | --------' | ' | | |' | | | 8, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [7, 6, 5, 4, 3, 2, 1, 0], 1, 40320, 40320 [7, 6, 5, 4, 3, 1, 1, 1], 2, 6720, 13440 [7, 6, 5, 4, 2, 2, 2, 0], 2, 6720, 13440 [7, 6, 5, 4, 2, 2, 1, 1], 4, 10080, 40320 [7, 6, 5, 3, 3, 3, 1, 0], 2, 6720, 13440 [7, 6, 5, 3, 3, 2, 2, 0], 4, 10080, 40320 [7, 6, 5, 3, 3, 2, 1, 1], 8, 10080, 80640 [7, 6, 5, 3, 2, 2, 2, 1], 14, 6720, 94080 [7, 6, 5, 2, 2, 2, 2, 2], 24, 336, 8064 [7, 6, 4, 4, 4, 2, 1, 0], 2, 6720, 13440 [7, 6, 4, 4, 4, 1, 1, 1], 4, 1120, 4480 [7, 6, 4, 4, 3, 3, 1, 0], 4, 10080, 40320 [7, 6, 4, 4, 3, 2, 2, 0], 8, 10080, 80640 [7, 6, 4, 4, 3, 2, 1, 1], 16, 10080, 161280 [7, 6, 4, 4, 2, 2, 2, 1], 28, 3360, 94080 [7, 6, 4, 3, 3, 3, 2, 0], 14, 6720, 94080 [7, 6, 4, 3, 3, 3, 1, 1], 28, 3360, 94080 [7, 6, 4, 3, 3, 2, 2, 1], 48, 10080, 483840 [7, 6, 4, 3, 2, 2, 2, 2], 80, 1680, 134400 [7, 6, 3, 3, 3, 3, 3, 0], 24, 336, 8064 [7, 6, 3, 3, 3, 3, 2, 1], 80, 1680, 134400 [7, 6, 3, 3, 3, 2, 2, 2], 132, 1120, 147840 [7, 5, 5, 5, 3, 2, 1, 0], 2, 6720, 13440 [7, 5, 5, 5, 3, 1, 1, 1], 4, 1120, 4480 [7, 5, 5, 5, 2, 2, 2, 0], 4, 1120, 4480 [7, 5, 5, 5, 2, 2, 1, 1], 8, 1680, 13440 [7, 5, 5, 4, 4, 2, 1, 0], 4, 10080, 40320 [7, 5, 5, 4, 4, 1, 1, 1], 8, 1680, 13440 [7, 5, 5, 4, 3, 3, 1, 0], 8, 10080, 80640 [7, 5, 5, 4, 3, 2, 2, 0], 16, 10080, 161280 [7, 5, 5, 4, 3, 2, 1, 1], 32, 10080, 322560 [7, 5, 5, 4, 2, 2, 2, 1], 56, 3360, 188160 [7, 5, 5, 3, 3, 3, 2, 0], 28, 3360, 94080 [7, 5, 5, 3, 3, 3, 1, 1], 56, 1680, 94080 [7, 5, 5, 3, 3, 2, 2, 1], 96, 5040, 483840 [7, 5, 5, 3, 2, 2, 2, 2], 160, 840, 134400 [7, 5, 4, 4, 4, 3, 1, 0], 14, 6720, 94080 [7, 5, 4, 4, 4, 2, 2, 0], 28, 3360, 94080 [7, 5, 4, 4, 4, 2, 1, 1], 56, 3360, 188160 [7, 5, 4, 4, 3, 3, 2, 0], 48, 10080, 483840 [7, 5, 4, 4, 3, 3, 1, 1], 96, 5040, 483840 [7, 5, 4, 4, 3, 2, 2, 1], 164, 10080, 1653120 [7, 5, 4, 4, 2, 2, 2, 2], 272, 840, 228480 [7, 5, 4, 3, 3, 3, 3, 0], 80, 1680, 134400 [7, 5, 4, 3, 3, 3, 2, 1], 266, 6720, 1787520 [7, 5, 4, 3, 3, 2, 2, 2], 436, 3360, 1464960 [7, 5, 3, 3, 3, 3, 3, 1], 424, 336, 142464 [7, 5, 3, 3, 3, 3, 2, 2], 688, 840, 577920 [7, 4, 4, 4, 4, 4, 1, 0], 24, 336, 8064 [7, 4, 4, 4, 4, 3, 2, 0], 80, 1680, 134400 [7, 4, 4, 4, 4, 3, 1, 1], 160, 840, 134400 [7, 4, 4, 4, 4, 2, 2, 1], 272, 840, 228480 [7, 4, 4, 4, 3, 3, 3, 0], 132, 1120, 147840 [7, 4, 4, 4, 3, 3, 2, 1], 436, 3360, 1464960 [7, 4, 4, 4, 3, 2, 2, 2], 712, 1120, 797440 [7, 4, 4, 3, 3, 3, 3, 1], 688, 840, 577920 [7, 4, 4, 3, 3, 3, 2, 2], 1112, 1680, 1868160 [7, 4, 3, 3, 3, 3, 3, 2], 1720, 336, 577920 [7, 3, 3, 3, 3, 3, 3, 3], 2640, 8, 21120 [6, 6, 6, 4, 3, 2, 1, 0], 2, 6720, 13440 [6, 6, 6, 4, 3, 1, 1, 1], 4, 1120, 4480 [6, 6, 6, 4, 2, 2, 2, 0], 4, 1120, 4480 [6, 6, 6, 4, 2, 2, 1, 1], 8, 1680, 13440 [6, 6, 6, 3, 3, 3, 1, 0], 4, 1120, 4480 [6, 6, 6, 3, 3, 2, 2, 0], 8, 1680, 13440 [6, 6, 6, 3, 3, 2, 1, 1], 16, 1680, 26880 [6, 6, 6, 3, 2, 2, 2, 1], 28, 1120, 31360 [6, 6, 6, 2, 2, 2, 2, 2], 48, 56, 2688 [6, 6, 5, 5, 3, 2, 1, 0], 4, 10080, 40320 [6, 6, 5, 5, 3, 1, 1, 1], 8, 1680, 13440 [6, 6, 5, 5, 2, 2, 2, 0], 8, 1680, 13440 [6, 6, 5, 5, 2, 2, 1, 1], 16, 2520, 40320 [6, 6, 5, 4, 4, 2, 1, 0], 8, 10080, 80640 [6, 6, 5, 4, 4, 1, 1, 1], 16, 1680, 26880 [6, 6, 5, 4, 3, 3, 1, 0], 16, 10080, 161280 [6, 6, 5, 4, 3, 2, 2, 0], 32, 10080, 322560 [6, 6, 5, 4, 3, 2, 1, 1], 64, 10080, 645120 [6, 6, 5, 4, 2, 2, 2, 1], 112, 3360, 376320 [6, 6, 5, 3, 3, 3, 2, 0], 56, 3360, 188160 [6, 6, 5, 3, 3, 3, 1, 1], 112, 1680, 188160 [6, 6, 5, 3, 3, 2, 2, 1], 192, 5040, 967680 [6, 6, 5, 3, 2, 2, 2, 2], 320, 840, 268800 [6, 6, 4, 4, 4, 3, 1, 0], 28, 3360, 94080 [6, 6, 4, 4, 4, 2, 2, 0], 56, 1680, 94080 [6, 6, 4, 4, 4, 2, 1, 1], 112, 1680, 188160 [6, 6, 4, 4, 3, 3, 2, 0], 96, 5040, 483840 [6, 6, 4, 4, 3, 3, 1, 1], 192, 2520, 483840 [6, 6, 4, 4, 3, 2, 2, 1], 328, 5040, 1653120 [6, 6, 4, 4, 2, 2, 2, 2], 544, 420, 228480 [6, 6, 4, 3, 3, 3, 3, 0], 160, 840, 134400 [6, 6, 4, 3, 3, 3, 2, 1], 532, 3360, 1787520 [6, 6, 4, 3, 3, 2, 2, 2], 872, 1680, 1464960 [6, 6, 3, 3, 3, 3, 3, 1], 848, 168, 142464 [6, 6, 3, 3, 3, 3, 2, 2], 1376, 420, 577920 [6, 5, 5, 5, 4, 2, 1, 0], 14, 6720, 94080 [6, 5, 5, 5, 4, 1, 1, 1], 28, 1120, 31360 [6, 5, 5, 5, 3, 3, 1, 0], 28, 3360, 94080 [6, 5, 5, 5, 3, 2, 2, 0], 56, 3360, 188160 [6, 5, 5, 5, 3, 2, 1, 1], 112, 3360, 376320 [6, 5, 5, 5, 2, 2, 2, 1], 196, 1120, 219520 [6, 5, 5, 4, 4, 3, 1, 0], 48, 10080, 483840 [6, 5, 5, 4, 4, 2, 2, 0], 96, 5040, 483840 [6, 5, 5, 4, 4, 2, 1, 1], 192, 5040, 967680 [6, 5, 5, 4, 3, 3, 2, 0], 164, 10080, 1653120 [6, 5, 5, 4, 3, 3, 1, 1], 328, 5040, 1653120 [6, 5, 5, 4, 3, 2, 2, 1], 560, 10080, 5644800 [6, 5, 5, 4, 2, 2, 2, 2], 928, 840, 779520 [6, 5, 5, 3, 3, 3, 3, 0], 272, 840, 228480 [6, 5, 5, 3, 3, 3, 2, 1], 904, 3360, 3037440 [6, 5, 5, 3, 3, 2, 2, 2], 1480, 1680, 2486400 [6, 5, 4, 4, 4, 4, 1, 0], 80, 1680, 134400 [6, 5, 4, 4, 4, 3, 2, 0], 266, 6720, 1787520 [6, 5, 4, 4, 4, 3, 1, 1], 532, 3360, 1787520 [6, 5, 4, 4, 4, 2, 2, 1], 904, 3360, 3037440 [6, 5, 4, 4, 3, 3, 3, 0], 436, 3360, 1464960 [6, 5, 4, 4, 3, 3, 2, 1], 1440, 10080, 14515200 [6, 5, 4, 4, 3, 2, 2, 2], 2348, 3360, 7889280 [6, 5, 4, 3, 3, 3, 3, 1], 2256, 1680, 3790080 [6, 5, 4, 3, 3, 3, 2, 2], 3640, 3360, 12230400 [6, 5, 3, 3, 3, 3, 3, 2], 5584, 336, 1876224 [6, 4, 4, 4, 4, 4, 2, 0], 424, 336, 142464 [6, 4, 4, 4, 4, 4, 1, 1], 848, 168, 142464 [6, 4, 4, 4, 4, 3, 3, 0], 688, 840, 577920 [6, 4, 4, 4, 4, 3, 2, 1], 2256, 1680, 3790080 [6, 4, 4, 4, 4, 2, 2, 2], 3664, 280, 1025920 [6, 4, 4, 4, 3, 3, 3, 1], 3504, 1120, 3924480 [6, 4, 4, 4, 3, 3, 2, 2], 5632, 1680, 9461760 [6, 4, 4, 3, 3, 3, 3, 2], 8576, 840, 7203840 [6, 4, 3, 3, 3, 3, 3, 3], 12960, 56, 725760 [5, 5, 5, 5, 5, 2, 1, 0], 24, 336, 8064 [5, 5, 5, 5, 5, 1, 1, 1], 48, 56, 2688 [5, 5, 5, 5, 4, 3, 1, 0], 80, 1680, 134400 [5, 5, 5, 5, 4, 2, 2, 0], 160, 840, 134400 [5, 5, 5, 5, 4, 2, 1, 1], 320, 840, 268800 [5, 5, 5, 5, 3, 3, 2, 0], 272, 840, 228480 [5, 5, 5, 5, 3, 3, 1, 1], 544, 420, 228480 [5, 5, 5, 5, 3, 2, 2, 1], 928, 840, 779520 [5, 5, 5, 5, 2, 2, 2, 2], 1536, 70, 107520 [5, 5, 5, 4, 4, 4, 1, 0], 132, 1120, 147840 [5, 5, 5, 4, 4, 3, 2, 0], 436, 3360, 1464960 [5, 5, 5, 4, 4, 3, 1, 1], 872, 1680, 1464960 [5, 5, 5, 4, 4, 2, 2, 1], 1480, 1680, 2486400 [5, 5, 5, 4, 3, 3, 3, 0], 712, 1120, 797440 [5, 5, 5, 4, 3, 3, 2, 1], 2348, 3360, 7889280 [5, 5, 5, 4, 3, 2, 2, 2], 3824, 1120, 4282880 [5, 5, 5, 3, 3, 3, 3, 1], 3664, 280, 1025920 [5, 5, 5, 3, 3, 3, 2, 2], 5904, 560, 3306240 [5, 5, 4, 4, 4, 4, 2, 0], 688, 840, 577920 [5, 5, 4, 4, 4, 4, 1, 1], 1376, 420, 577920 [5, 5, 4, 4, 4, 3, 3, 0], 1112, 1680, 1868160 [5, 5, 4, 4, 4, 3, 2, 1], 3640, 3360, 12230400 [5, 5, 4, 4, 4, 2, 2, 2], 5904, 560, 3306240 [5, 5, 4, 4, 3, 3, 3, 1], 5632, 1680, 9461760 [5, 5, 4, 4, 3, 3, 2, 2], 9040, 2520, 22780800 [5, 5, 4, 3, 3, 3, 3, 2], 13712, 840, 11518080 [5, 5, 3, 3, 3, 3, 3, 3], 20640, 28, 577920 [5, 4, 4, 4, 4, 4, 3, 0], 1720, 336, 577920 [5, 4, 4, 4, 4, 4, 2, 1], 5584, 336, 1876224 [5, 4, 4, 4, 4, 3, 3, 1], 8576, 840, 7203840 [5, 4, 4, 4, 4, 3, 2, 2], 13712, 840, 11518080 [5, 4, 4, 4, 3, 3, 3, 2], 20676, 1120, 23157120 [5, 4, 4, 3, 3, 3, 3, 3], 30960, 168, 5201280 [4, 4, 4, 4, 4, 4, 4, 0], 2640, 8, 21120 [4, 4, 4, 4, 4, 4, 3, 1], 12960, 56, 725760 [4, 4, 4, 4, 4, 4, 2, 2], 20640, 28, 577920 [4, 4, 4, 4, 4, 3, 3, 2], 30960, 168, 5201280 [4, 4, 4, 4, 3, 3, 3, 3], 46144, 70, 3230080 Sum of numbers in last column=, 268435456 The most probable vector is, [5, 4, 4, 4, 3, 3, 3, 2], which arises in, 23157120, out of 2 to the power, 28, that equals, 268435456, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 8, players The sole champion score-vector is, [5, 4, 4, 4, 3, 3, 3, 2] the corresponding record is, 23157120 ----------------------------------------------------------------------------\ ----- 9 / 9 \ --------' | --------' | ' | | |' | | | 9, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [8, 7, 6, 5, 4, 3, 2, 1, 0], 1, 362880, 362880 [8, 7, 6, 5, 4, 3, 1, 1, 1], 2, 60480, 120960 [8, 7, 6, 5, 4, 2, 2, 2, 0], 2, 60480, 120960 [8, 7, 6, 5, 4, 2, 2, 1, 1], 4, 90720, 362880 [8, 7, 6, 5, 3, 3, 3, 1, 0], 2, 60480, 120960 [8, 7, 6, 5, 3, 3, 2, 2, 0], 4, 90720, 362880 [8, 7, 6, 5, 3, 3, 2, 1, 1], 8, 90720, 725760 [8, 7, 6, 5, 3, 2, 2, 2, 1], 14, 60480, 846720 [8, 7, 6, 5, 2, 2, 2, 2, 2], 24, 3024, 72576 [8, 7, 6, 4, 4, 4, 2, 1, 0], 2, 60480, 120960 [8, 7, 6, 4, 4, 4, 1, 1, 1], 4, 10080, 40320 [8, 7, 6, 4, 4, 3, 3, 1, 0], 4, 90720, 362880 [8, 7, 6, 4, 4, 3, 2, 2, 0], 8, 90720, 725760 [8, 7, 6, 4, 4, 3, 2, 1, 1], 16, 90720, 1451520 [8, 7, 6, 4, 4, 2, 2, 2, 1], 28, 30240, 846720 [8, 7, 6, 4, 3, 3, 3, 2, 0], 14, 60480, 846720 [8, 7, 6, 4, 3, 3, 3, 1, 1], 28, 30240, 846720 [8, 7, 6, 4, 3, 3, 2, 2, 1], 48, 90720, 4354560 [8, 7, 6, 4, 3, 2, 2, 2, 2], 80, 15120, 1209600 [8, 7, 6, 3, 3, 3, 3, 3, 0], 24, 3024, 72576 [8, 7, 6, 3, 3, 3, 3, 2, 1], 80, 15120, 1209600 [8, 7, 6, 3, 3, 3, 2, 2, 2], 132, 10080, 1330560 [8, 7, 5, 5, 5, 3, 2, 1, 0], 2, 60480, 120960 [8, 7, 5, 5, 5, 3, 1, 1, 1], 4, 10080, 40320 [8, 7, 5, 5, 5, 2, 2, 2, 0], 4, 10080, 40320 [8, 7, 5, 5, 5, 2, 2, 1, 1], 8, 15120, 120960 [8, 7, 5, 5, 4, 4, 2, 1, 0], 4, 90720, 362880 [8, 7, 5, 5, 4, 4, 1, 1, 1], 8, 15120, 120960 [8, 7, 5, 5, 4, 3, 3, 1, 0], 8, 90720, 725760 [8, 7, 5, 5, 4, 3, 2, 2, 0], 16, 90720, 1451520 [8, 7, 5, 5, 4, 3, 2, 1, 1], 32, 90720, 2903040 [8, 7, 5, 5, 4, 2, 2, 2, 1], 56, 30240, 1693440 [8, 7, 5, 5, 3, 3, 3, 2, 0], 28, 30240, 846720 [8, 7, 5, 5, 3, 3, 3, 1, 1], 56, 15120, 846720 [8, 7, 5, 5, 3, 3, 2, 2, 1], 96, 45360, 4354560 [8, 7, 5, 5, 3, 2, 2, 2, 2], 160, 7560, 1209600 [8, 7, 5, 4, 4, 4, 3, 1, 0], 14, 60480, 846720 [8, 7, 5, 4, 4, 4, 2, 2, 0], 28, 30240, 846720 [8, 7, 5, 4, 4, 4, 2, 1, 1], 56, 30240, 1693440 [8, 7, 5, 4, 4, 3, 3, 2, 0], 48, 90720, 4354560 [8, 7, 5, 4, 4, 3, 3, 1, 1], 96, 45360, 4354560 [8, 7, 5, 4, 4, 3, 2, 2, 1], 164, 90720, 14878080 [8, 7, 5, 4, 4, 2, 2, 2, 2], 272, 7560, 2056320 [8, 7, 5, 4, 3, 3, 3, 3, 0], 80, 15120, 1209600 [8, 7, 5, 4, 3, 3, 3, 2, 1], 266, 60480, 16087680 [8, 7, 5, 4, 3, 3, 2, 2, 2], 436, 30240, 13184640 [8, 7, 5, 3, 3, 3, 3, 3, 1], 424, 3024, 1282176 [8, 7, 5, 3, 3, 3, 3, 2, 2], 688, 7560, 5201280 [8, 7, 4, 4, 4, 4, 4, 1, 0], 24, 3024, 72576 [8, 7, 4, 4, 4, 4, 3, 2, 0], 80, 15120, 1209600 [8, 7, 4, 4, 4, 4, 3, 1, 1], 160, 7560, 1209600 [8, 7, 4, 4, 4, 4, 2, 2, 1], 272, 7560, 2056320 [8, 7, 4, 4, 4, 3, 3, 3, 0], 132, 10080, 1330560 [8, 7, 4, 4, 4, 3, 3, 2, 1], 436, 30240, 13184640 [8, 7, 4, 4, 4, 3, 2, 2, 2], 712, 10080, 7176960 [8, 7, 4, 4, 3, 3, 3, 3, 1], 688, 7560, 5201280 [8, 7, 4, 4, 3, 3, 3, 2, 2], 1112, 15120, 16813440 [8, 7, 4, 3, 3, 3, 3, 3, 2], 1720, 3024, 5201280 [8, 7, 3, 3, 3, 3, 3, 3, 3], 2640, 72, 190080 [8, 6, 6, 6, 4, 3, 2, 1, 0], 2, 60480, 120960 [8, 6, 6, 6, 4, 3, 1, 1, 1], 4, 10080, 40320 [8, 6, 6, 6, 4, 2, 2, 2, 0], 4, 10080, 40320 [8, 6, 6, 6, 4, 2, 2, 1, 1], 8, 15120, 120960 [8, 6, 6, 6, 3, 3, 3, 1, 0], 4, 10080, 40320 [8, 6, 6, 6, 3, 3, 2, 2, 0], 8, 15120, 120960 [8, 6, 6, 6, 3, 3, 2, 1, 1], 16, 15120, 241920 [8, 6, 6, 6, 3, 2, 2, 2, 1], 28, 10080, 282240 [8, 6, 6, 6, 2, 2, 2, 2, 2], 48, 504, 24192 [8, 6, 6, 5, 5, 3, 2, 1, 0], 4, 90720, 362880 [8, 6, 6, 5, 5, 3, 1, 1, 1], 8, 15120, 120960 [8, 6, 6, 5, 5, 2, 2, 2, 0], 8, 15120, 120960 [8, 6, 6, 5, 5, 2, 2, 1, 1], 16, 22680, 362880 [8, 6, 6, 5, 4, 4, 2, 1, 0], 8, 90720, 725760 [8, 6, 6, 5, 4, 4, 1, 1, 1], 16, 15120, 241920 [8, 6, 6, 5, 4, 3, 3, 1, 0], 16, 90720, 1451520 [8, 6, 6, 5, 4, 3, 2, 2, 0], 32, 90720, 2903040 [8, 6, 6, 5, 4, 3, 2, 1, 1], 64, 90720, 5806080 [8, 6, 6, 5, 4, 2, 2, 2, 1], 112, 30240, 3386880 [8, 6, 6, 5, 3, 3, 3, 2, 0], 56, 30240, 1693440 [8, 6, 6, 5, 3, 3, 3, 1, 1], 112, 15120, 1693440 [8, 6, 6, 5, 3, 3, 2, 2, 1], 192, 45360, 8709120 [8, 6, 6, 5, 3, 2, 2, 2, 2], 320, 7560, 2419200 [8, 6, 6, 4, 4, 4, 3, 1, 0], 28, 30240, 846720 [8, 6, 6, 4, 4, 4, 2, 2, 0], 56, 15120, 846720 [8, 6, 6, 4, 4, 4, 2, 1, 1], 112, 15120, 1693440 [8, 6, 6, 4, 4, 3, 3, 2, 0], 96, 45360, 4354560 [8, 6, 6, 4, 4, 3, 3, 1, 1], 192, 22680, 4354560 [8, 6, 6, 4, 4, 3, 2, 2, 1], 328, 45360, 14878080 [8, 6, 6, 4, 4, 2, 2, 2, 2], 544, 3780, 2056320 [8, 6, 6, 4, 3, 3, 3, 3, 0], 160, 7560, 1209600 [8, 6, 6, 4, 3, 3, 3, 2, 1], 532, 30240, 16087680 [8, 6, 6, 4, 3, 3, 2, 2, 2], 872, 15120, 13184640 [8, 6, 6, 3, 3, 3, 3, 3, 1], 848, 1512, 1282176 [8, 6, 6, 3, 3, 3, 3, 2, 2], 1376, 3780, 5201280 [8, 6, 5, 5, 5, 4, 2, 1, 0], 14, 60480, 846720 [8, 6, 5, 5, 5, 4, 1, 1, 1], 28, 10080, 282240 [8, 6, 5, 5, 5, 3, 3, 1, 0], 28, 30240, 846720 [8, 6, 5, 5, 5, 3, 2, 2, 0], 56, 30240, 1693440 [8, 6, 5, 5, 5, 3, 2, 1, 1], 112, 30240, 3386880 [8, 6, 5, 5, 5, 2, 2, 2, 1], 196, 10080, 1975680 [8, 6, 5, 5, 4, 4, 3, 1, 0], 48, 90720, 4354560 [8, 6, 5, 5, 4, 4, 2, 2, 0], 96, 45360, 4354560 [8, 6, 5, 5, 4, 4, 2, 1, 1], 192, 45360, 8709120 [8, 6, 5, 5, 4, 3, 3, 2, 0], 164, 90720, 14878080 [8, 6, 5, 5, 4, 3, 3, 1, 1], 328, 45360, 14878080 [8, 6, 5, 5, 4, 3, 2, 2, 1], 560, 90720, 50803200 [8, 6, 5, 5, 4, 2, 2, 2, 2], 928, 7560, 7015680 [8, 6, 5, 5, 3, 3, 3, 3, 0], 272, 7560, 2056320 [8, 6, 5, 5, 3, 3, 3, 2, 1], 904, 30240, 27336960 [8, 6, 5, 5, 3, 3, 2, 2, 2], 1480, 15120, 22377600 [8, 6, 5, 4, 4, 4, 4, 1, 0], 80, 15120, 1209600 [8, 6, 5, 4, 4, 4, 3, 2, 0], 266, 60480, 16087680 [8, 6, 5, 4, 4, 4, 3, 1, 1], 532, 30240, 16087680 [8, 6, 5, 4, 4, 4, 2, 2, 1], 904, 30240, 27336960 [8, 6, 5, 4, 4, 3, 3, 3, 0], 436, 30240, 13184640 [8, 6, 5, 4, 4, 3, 3, 2, 1], 1440, 90720, 130636800 [8, 6, 5, 4, 4, 3, 2, 2, 2], 2348, 30240, 71003520 [8, 6, 5, 4, 3, 3, 3, 3, 1], 2256, 15120, 34110720 [8, 6, 5, 4, 3, 3, 3, 2, 2], 3640, 30240, 110073600 [8, 6, 5, 3, 3, 3, 3, 3, 2], 5584, 3024, 16886016 [8, 6, 4, 4, 4, 4, 4, 2, 0], 424, 3024, 1282176 [8, 6, 4, 4, 4, 4, 4, 1, 1], 848, 1512, 1282176 [8, 6, 4, 4, 4, 4, 3, 3, 0], 688, 7560, 5201280 [8, 6, 4, 4, 4, 4, 3, 2, 1], 2256, 15120, 34110720 [8, 6, 4, 4, 4, 4, 2, 2, 2], 3664, 2520, 9233280 [8, 6, 4, 4, 4, 3, 3, 3, 1], 3504, 10080, 35320320 [8, 6, 4, 4, 4, 3, 3, 2, 2], 5632, 15120, 85155840 [8, 6, 4, 4, 3, 3, 3, 3, 2], 8576, 7560, 64834560 [8, 6, 4, 3, 3, 3, 3, 3, 3], 12960, 504, 6531840 [8, 5, 5, 5, 5, 5, 2, 1, 0], 24, 3024, 72576 [8, 5, 5, 5, 5, 5, 1, 1, 1], 48, 504, 24192 [8, 5, 5, 5, 5, 4, 3, 1, 0], 80, 15120, 1209600 [8, 5, 5, 5, 5, 4, 2, 2, 0], 160, 7560, 1209600 [8, 5, 5, 5, 5, 4, 2, 1, 1], 320, 7560, 2419200 [8, 5, 5, 5, 5, 3, 3, 2, 0], 272, 7560, 2056320 [8, 5, 5, 5, 5, 3, 3, 1, 1], 544, 3780, 2056320 [8, 5, 5, 5, 5, 3, 2, 2, 1], 928, 7560, 7015680 [8, 5, 5, 5, 5, 2, 2, 2, 2], 1536, 630, 967680 [8, 5, 5, 5, 4, 4, 4, 1, 0], 132, 10080, 1330560 [8, 5, 5, 5, 4, 4, 3, 2, 0], 436, 30240, 13184640 [8, 5, 5, 5, 4, 4, 3, 1, 1], 872, 15120, 13184640 [8, 5, 5, 5, 4, 4, 2, 2, 1], 1480, 15120, 22377600 [8, 5, 5, 5, 4, 3, 3, 3, 0], 712, 10080, 7176960 [8, 5, 5, 5, 4, 3, 3, 2, 1], 2348, 30240, 71003520 [8, 5, 5, 5, 4, 3, 2, 2, 2], 3824, 10080, 38545920 [8, 5, 5, 5, 3, 3, 3, 3, 1], 3664, 2520, 9233280 [8, 5, 5, 5, 3, 3, 3, 2, 2], 5904, 5040, 29756160 [8, 5, 5, 4, 4, 4, 4, 2, 0], 688, 7560, 5201280 [8, 5, 5, 4, 4, 4, 4, 1, 1], 1376, 3780, 5201280 [8, 5, 5, 4, 4, 4, 3, 3, 0], 1112, 15120, 16813440 [8, 5, 5, 4, 4, 4, 3, 2, 1], 3640, 30240, 110073600 [8, 5, 5, 4, 4, 4, 2, 2, 2], 5904, 5040, 29756160 [8, 5, 5, 4, 4, 3, 3, 3, 1], 5632, 15120, 85155840 [8, 5, 5, 4, 4, 3, 3, 2, 2], 9040, 22680, 205027200 [8, 5, 5, 4, 3, 3, 3, 3, 2], 13712, 7560, 103662720 [8, 5, 5, 3, 3, 3, 3, 3, 3], 20640, 252, 5201280 [8, 5, 4, 4, 4, 4, 4, 3, 0], 1720, 3024, 5201280 [8, 5, 4, 4, 4, 4, 4, 2, 1], 5584, 3024, 16886016 [8, 5, 4, 4, 4, 4, 3, 3, 1], 8576, 7560, 64834560 [8, 5, 4, 4, 4, 4, 3, 2, 2], 13712, 7560, 103662720 [8, 5, 4, 4, 4, 3, 3, 3, 2], 20676, 10080, 208414080 [8, 5, 4, 4, 3, 3, 3, 3, 3], 30960, 1512, 46811520 [8, 4, 4, 4, 4, 4, 4, 4, 0], 2640, 72, 190080 [8, 4, 4, 4, 4, 4, 4, 3, 1], 12960, 504, 6531840 [8, 4, 4, 4, 4, 4, 4, 2, 2], 20640, 252, 5201280 [8, 4, 4, 4, 4, 4, 3, 3, 2], 30960, 1512, 46811520 [8, 4, 4, 4, 4, 3, 3, 3, 3], 46144, 630, 29070720 [7, 7, 7, 5, 4, 3, 2, 1, 0], 2, 60480, 120960 [7, 7, 7, 5, 4, 3, 1, 1, 1], 4, 10080, 40320 [7, 7, 7, 5, 4, 2, 2, 2, 0], 4, 10080, 40320 [7, 7, 7, 5, 4, 2, 2, 1, 1], 8, 15120, 120960 [7, 7, 7, 5, 3, 3, 3, 1, 0], 4, 10080, 40320 [7, 7, 7, 5, 3, 3, 2, 2, 0], 8, 15120, 120960 [7, 7, 7, 5, 3, 3, 2, 1, 1], 16, 15120, 241920 [7, 7, 7, 5, 3, 2, 2, 2, 1], 28, 10080, 282240 [7, 7, 7, 5, 2, 2, 2, 2, 2], 48, 504, 24192 [7, 7, 7, 4, 4, 4, 2, 1, 0], 4, 10080, 40320 [7, 7, 7, 4, 4, 4, 1, 1, 1], 8, 1680, 13440 [7, 7, 7, 4, 4, 3, 3, 1, 0], 8, 15120, 120960 [7, 7, 7, 4, 4, 3, 2, 2, 0], 16, 15120, 241920 [7, 7, 7, 4, 4, 3, 2, 1, 1], 32, 15120, 483840 [7, 7, 7, 4, 4, 2, 2, 2, 1], 56, 5040, 282240 [7, 7, 7, 4, 3, 3, 3, 2, 0], 28, 10080, 282240 [7, 7, 7, 4, 3, 3, 3, 1, 1], 56, 5040, 282240 [7, 7, 7, 4, 3, 3, 2, 2, 1], 96, 15120, 1451520 [7, 7, 7, 4, 3, 2, 2, 2, 2], 160, 2520, 403200 [7, 7, 7, 3, 3, 3, 3, 3, 0], 48, 504, 24192 [7, 7, 7, 3, 3, 3, 3, 2, 1], 160, 2520, 403200 [7, 7, 7, 3, 3, 3, 2, 2, 2], 264, 1680, 443520 [7, 7, 6, 6, 4, 3, 2, 1, 0], 4, 90720, 362880 [7, 7, 6, 6, 4, 3, 1, 1, 1], 8, 15120, 120960 [7, 7, 6, 6, 4, 2, 2, 2, 0], 8, 15120, 120960 [7, 7, 6, 6, 4, 2, 2, 1, 1], 16, 22680, 362880 [7, 7, 6, 6, 3, 3, 3, 1, 0], 8, 15120, 120960 [7, 7, 6, 6, 3, 3, 2, 2, 0], 16, 22680, 362880 [7, 7, 6, 6, 3, 3, 2, 1, 1], 32, 22680, 725760 [7, 7, 6, 6, 3, 2, 2, 2, 1], 56, 15120, 846720 [7, 7, 6, 6, 2, 2, 2, 2, 2], 96, 756, 72576 [7, 7, 6, 5, 5, 3, 2, 1, 0], 8, 90720, 725760 [7, 7, 6, 5, 5, 3, 1, 1, 1], 16, 15120, 241920 [7, 7, 6, 5, 5, 2, 2, 2, 0], 16, 15120, 241920 [7, 7, 6, 5, 5, 2, 2, 1, 1], 32, 22680, 725760 [7, 7, 6, 5, 4, 4, 2, 1, 0], 16, 90720, 1451520 [7, 7, 6, 5, 4, 4, 1, 1, 1], 32, 15120, 483840 [7, 7, 6, 5, 4, 3, 3, 1, 0], 32, 90720, 2903040 [7, 7, 6, 5, 4, 3, 2, 2, 0], 64, 90720, 5806080 [7, 7, 6, 5, 4, 3, 2, 1, 1], 128, 90720, 11612160 [7, 7, 6, 5, 4, 2, 2, 2, 1], 224, 30240, 6773760 [7, 7, 6, 5, 3, 3, 3, 2, 0], 112, 30240, 3386880 [7, 7, 6, 5, 3, 3, 3, 1, 1], 224, 15120, 3386880 [7, 7, 6, 5, 3, 3, 2, 2, 1], 384, 45360, 17418240 [7, 7, 6, 5, 3, 2, 2, 2, 2], 640, 7560, 4838400 [7, 7, 6, 4, 4, 4, 3, 1, 0], 56, 30240, 1693440 [7, 7, 6, 4, 4, 4, 2, 2, 0], 112, 15120, 1693440 [7, 7, 6, 4, 4, 4, 2, 1, 1], 224, 15120, 3386880 [7, 7, 6, 4, 4, 3, 3, 2, 0], 192, 45360, 8709120 [7, 7, 6, 4, 4, 3, 3, 1, 1], 384, 22680, 8709120 [7, 7, 6, 4, 4, 3, 2, 2, 1], 656, 45360, 29756160 [7, 7, 6, 4, 4, 2, 2, 2, 2], 1088, 3780, 4112640 [7, 7, 6, 4, 3, 3, 3, 3, 0], 320, 7560, 2419200 [7, 7, 6, 4, 3, 3, 3, 2, 1], 1064, 30240, 32175360 [7, 7, 6, 4, 3, 3, 2, 2, 2], 1744, 15120, 26369280 [7, 7, 6, 3, 3, 3, 3, 3, 1], 1696, 1512, 2564352 [7, 7, 6, 3, 3, 3, 3, 2, 2], 2752, 3780, 10402560 [7, 7, 5, 5, 5, 4, 2, 1, 0], 28, 30240, 846720 [7, 7, 5, 5, 5, 4, 1, 1, 1], 56, 5040, 282240 [7, 7, 5, 5, 5, 3, 3, 1, 0], 56, 15120, 846720 [7, 7, 5, 5, 5, 3, 2, 2, 0], 112, 15120, 1693440 [7, 7, 5, 5, 5, 3, 2, 1, 1], 224, 15120, 3386880 [7, 7, 5, 5, 5, 2, 2, 2, 1], 392, 5040, 1975680 [7, 7, 5, 5, 4, 4, 3, 1, 0], 96, 45360, 4354560 [7, 7, 5, 5, 4, 4, 2, 2, 0], 192, 22680, 4354560 [7, 7, 5, 5, 4, 4, 2, 1, 1], 384, 22680, 8709120 [7, 7, 5, 5, 4, 3, 3, 2, 0], 328, 45360, 14878080 [7, 7, 5, 5, 4, 3, 3, 1, 1], 656, 22680, 14878080 [7, 7, 5, 5, 4, 3, 2, 2, 1], 1120, 45360, 50803200 [7, 7, 5, 5, 4, 2, 2, 2, 2], 1856, 3780, 7015680 [7, 7, 5, 5, 3, 3, 3, 3, 0], 544, 3780, 2056320 [7, 7, 5, 5, 3, 3, 3, 2, 1], 1808, 15120, 27336960 [7, 7, 5, 5, 3, 3, 2, 2, 2], 2960, 7560, 22377600 [7, 7, 5, 4, 4, 4, 4, 1, 0], 160, 7560, 1209600 [7, 7, 5, 4, 4, 4, 3, 2, 0], 532, 30240, 16087680 [7, 7, 5, 4, 4, 4, 3, 1, 1], 1064, 15120, 16087680 [7, 7, 5, 4, 4, 4, 2, 2, 1], 1808, 15120, 27336960 [7, 7, 5, 4, 4, 3, 3, 3, 0], 872, 15120, 13184640 [7, 7, 5, 4, 4, 3, 3, 2, 1], 2880, 45360, 130636800 [7, 7, 5, 4, 4, 3, 2, 2, 2], 4696, 15120, 71003520 [7, 7, 5, 4, 3, 3, 3, 3, 1], 4512, 7560, 34110720 [7, 7, 5, 4, 3, 3, 3, 2, 2], 7280, 15120, 110073600 [7, 7, 5, 3, 3, 3, 3, 3, 2], 11168, 1512, 16886016 [7, 7, 4, 4, 4, 4, 4, 2, 0], 848, 1512, 1282176 [7, 7, 4, 4, 4, 4, 4, 1, 1], 1696, 756, 1282176 [7, 7, 4, 4, 4, 4, 3, 3, 0], 1376, 3780, 5201280 [7, 7, 4, 4, 4, 4, 3, 2, 1], 4512, 7560, 34110720 [7, 7, 4, 4, 4, 4, 2, 2, 2], 7328, 1260, 9233280 [7, 7, 4, 4, 4, 3, 3, 3, 1], 7008, 5040, 35320320 [7, 7, 4, 4, 4, 3, 3, 2, 2], 11264, 7560, 85155840 [7, 7, 4, 4, 3, 3, 3, 3, 2], 17152, 3780, 64834560 [7, 7, 4, 3, 3, 3, 3, 3, 3], 25920, 252, 6531840 [7, 6, 6, 6, 5, 3, 2, 1, 0], 14, 60480, 846720 [7, 6, 6, 6, 5, 3, 1, 1, 1], 28, 10080, 282240 [7, 6, 6, 6, 5, 2, 2, 2, 0], 28, 10080, 282240 [7, 6, 6, 6, 5, 2, 2, 1, 1], 56, 15120, 846720 [7, 6, 6, 6, 4, 4, 2, 1, 0], 28, 30240, 846720 [7, 6, 6, 6, 4, 4, 1, 1, 1], 56, 5040, 282240 [7, 6, 6, 6, 4, 3, 3, 1, 0], 56, 30240, 1693440 [7, 6, 6, 6, 4, 3, 2, 2, 0], 112, 30240, 3386880 [7, 6, 6, 6, 4, 3, 2, 1, 1], 224, 30240, 6773760 [7, 6, 6, 6, 4, 2, 2, 2, 1], 392, 10080, 3951360 [7, 6, 6, 6, 3, 3, 3, 2, 0], 196, 10080, 1975680 [7, 6, 6, 6, 3, 3, 3, 1, 1], 392, 5040, 1975680 [7, 6, 6, 6, 3, 3, 2, 2, 1], 672, 15120, 10160640 [7, 6, 6, 6, 3, 2, 2, 2, 2], 1120, 2520, 2822400 [7, 6, 6, 5, 5, 4, 2, 1, 0], 48, 90720, 4354560 [7, 6, 6, 5, 5, 4, 1, 1, 1], 96, 15120, 1451520 [7, 6, 6, 5, 5, 3, 3, 1, 0], 96, 45360, 4354560 [7, 6, 6, 5, 5, 3, 2, 2, 0], 192, 45360, 8709120 [7, 6, 6, 5, 5, 3, 2, 1, 1], 384, 45360, 17418240 [7, 6, 6, 5, 5, 2, 2, 2, 1], 672, 15120, 10160640 [7, 6, 6, 5, 4, 4, 3, 1, 0], 164, 90720, 14878080 [7, 6, 6, 5, 4, 4, 2, 2, 0], 328, 45360, 14878080 [7, 6, 6, 5, 4, 4, 2, 1, 1], 656, 45360, 29756160 [7, 6, 6, 5, 4, 3, 3, 2, 0], 560, 90720, 50803200 [7, 6, 6, 5, 4, 3, 3, 1, 1], 1120, 45360, 50803200 [7, 6, 6, 5, 4, 3, 2, 2, 1], 1912, 90720, 173456640 [7, 6, 6, 5, 4, 2, 2, 2, 2], 3168, 7560, 23950080 [7, 6, 6, 5, 3, 3, 3, 3, 0], 928, 7560, 7015680 [7, 6, 6, 5, 3, 3, 3, 2, 1], 3084, 30240, 93260160 [7, 6, 6, 5, 3, 3, 2, 2, 2], 5048, 15120, 76325760 [7, 6, 6, 4, 4, 4, 4, 1, 0], 272, 7560, 2056320 [7, 6, 6, 4, 4, 4, 3, 2, 0], 904, 30240, 27336960 [7, 6, 6, 4, 4, 4, 3, 1, 1], 1808, 15120, 27336960 [7, 6, 6, 4, 4, 4, 2, 2, 1], 3072, 15120, 46448640 [7, 6, 6, 4, 4, 3, 3, 3, 0], 1480, 15120, 22377600 [7, 6, 6, 4, 4, 3, 3, 2, 1], 4888, 45360, 221719680 [7, 6, 6, 4, 4, 3, 2, 2, 2], 7968, 15120, 120476160 [7, 6, 6, 4, 3, 3, 3, 3, 1], 7648, 7560, 57818880 [7, 6, 6, 4, 3, 3, 3, 2, 2], 12336, 15120, 186520320 [7, 6, 6, 3, 3, 3, 3, 3, 2], 18896, 1512, 28570752 [7, 6, 5, 5, 5, 5, 2, 1, 0], 80, 15120, 1209600 [7, 6, 5, 5, 5, 5, 1, 1, 1], 160, 2520, 403200 [7, 6, 5, 5, 5, 4, 3, 1, 0], 266, 60480, 16087680 [7, 6, 5, 5, 5, 4, 2, 2, 0], 532, 30240, 16087680 [7, 6, 5, 5, 5, 4, 2, 1, 1], 1064, 30240, 32175360 [7, 6, 5, 5, 5, 3, 3, 2, 0], 904, 30240, 27336960 [7, 6, 5, 5, 5, 3, 3, 1, 1], 1808, 15120, 27336960 [7, 6, 5, 5, 5, 3, 2, 2, 1], 3084, 30240, 93260160 [7, 6, 5, 5, 5, 2, 2, 2, 2], 5104, 2520, 12862080 [7, 6, 5, 5, 4, 4, 4, 1, 0], 436, 30240, 13184640 [7, 6, 5, 5, 4, 4, 3, 2, 0], 1440, 90720, 130636800 [7, 6, 5, 5, 4, 4, 3, 1, 1], 2880, 45360, 130636800 [7, 6, 5, 5, 4, 4, 2, 2, 1], 4888, 45360, 221719680 [7, 6, 5, 5, 4, 3, 3, 3, 0], 2348, 30240, 71003520 [7, 6, 5, 5, 4, 3, 3, 2, 1], 7744, 90720, 702535680 [7, 6, 5, 5, 4, 3, 2, 2, 2], 12608, 30240, 381265920 [7, 6, 5, 5, 3, 3, 3, 3, 1], 12064, 7560, 91203840 [7, 6, 5, 5, 3, 3, 3, 2, 2], 19432, 15120, 293811840 [7, 6, 5, 4, 4, 4, 4, 2, 0], 2256, 15120, 34110720 [7, 6, 5, 4, 4, 4, 4, 1, 1], 4512, 7560, 34110720 [7, 6, 5, 4, 4, 4, 3, 3, 0], 3640, 30240, 110073600 [7, 6, 5, 4, 4, 4, 3, 2, 1], 11918, 60480, 720800640 [7, 6, 5, 4, 4, 4, 2, 2, 2], 19324, 10080, 194785920 [7, 6, 5, 4, 4, 3, 3, 3, 1], 18404, 30240, 556536960 [7, 6, 5, 4, 4, 3, 3, 2, 2], 29528, 45360, 1339390080 [7, 6, 5, 4, 3, 3, 3, 3, 2], 44688, 15120, 675682560 [7, 6, 5, 3, 3, 3, 3, 3, 3], 67104, 504, 33820416 [7, 6, 4, 4, 4, 4, 4, 3, 0], 5584, 3024, 16886016 [7, 6, 4, 4, 4, 4, 4, 2, 1], 18136, 3024, 54843264 [7, 6, 4, 4, 4, 4, 3, 3, 1], 27792, 7560, 210107520 [7, 6, 4, 4, 4, 4, 3, 2, 2], 44416, 7560, 335784960 [7, 6, 4, 4, 4, 3, 3, 3, 2], 66804, 10080, 673384320 [7, 6, 4, 4, 3, 3, 3, 3, 3], 99760, 1512, 150837120 [7, 5, 5, 5, 5, 5, 3, 1, 0], 424, 3024, 1282176 [7, 5, 5, 5, 5, 5, 2, 2, 0], 848, 1512, 1282176 [7, 5, 5, 5, 5, 5, 2, 1, 1], 1696, 1512, 2564352 [7, 5, 5, 5, 5, 4, 4, 1, 0], 688, 7560, 5201280 [7, 5, 5, 5, 5, 4, 3, 2, 0], 2256, 15120, 34110720 [7, 5, 5, 5, 5, 4, 3, 1, 1], 4512, 7560, 34110720 [7, 5, 5, 5, 5, 4, 2, 2, 1], 7648, 7560, 57818880 [7, 5, 5, 5, 5, 3, 3, 3, 0], 3664, 2520, 9233280 [7, 5, 5, 5, 5, 3, 3, 2, 1], 12064, 7560, 91203840 [7, 5, 5, 5, 5, 3, 2, 2, 2], 19616, 2520, 49432320 [7, 5, 5, 5, 4, 4, 4, 2, 0], 3504, 10080, 35320320 [7, 5, 5, 5, 4, 4, 4, 1, 1], 7008, 5040, 35320320 [7, 5, 5, 5, 4, 4, 3, 3, 0], 5632, 15120, 85155840 [7, 5, 5, 5, 4, 4, 3, 2, 1], 18404, 30240, 556536960 [7, 5, 5, 5, 4, 4, 2, 2, 2], 29800, 5040, 150192000 [7, 5, 5, 5, 4, 3, 3, 3, 1], 28316, 10080, 285425280 [7, 5, 5, 5, 4, 3, 3, 2, 2], 45368, 15120, 685964160 [7, 5, 5, 5, 3, 3, 3, 3, 2], 68416, 2520, 172408320 [7, 5, 5, 4, 4, 4, 4, 3, 0], 8576, 7560, 64834560 [7, 5, 5, 4, 4, 4, 4, 2, 1], 27792, 7560, 210107520 [7, 5, 5, 4, 4, 4, 3, 3, 1], 42440, 15120, 641692800 [7, 5, 5, 4, 4, 4, 3, 2, 2], 67728, 15120, 1024047360 [7, 5, 5, 4, 4, 3, 3, 3, 2], 101528, 15120, 1535103360 [7, 5, 5, 4, 3, 3, 3, 3, 3], 151120, 1512, 228493440 [7, 5, 4, 4, 4, 4, 4, 4, 0], 12960, 504, 6531840 [7, 5, 4, 4, 4, 4, 4, 3, 1], 63144, 3024, 190947456 [7, 5, 4, 4, 4, 4, 4, 2, 2], 100368, 1512, 151756416 [7, 5, 4, 4, 4, 4, 3, 3, 2], 149664, 7560, 1131459840 [7, 5, 4, 4, 4, 3, 3, 3, 3], 221728, 2520, 558754560 [7, 4, 4, 4, 4, 4, 4, 4, 1], 93360, 72, 6721920 [7, 4, 4, 4, 4, 4, 4, 3, 2], 219360, 504, 110557440 [7, 4, 4, 4, 4, 4, 3, 3, 3], 323600, 504, 163094400 [6, 6, 6, 6, 6, 3, 2, 1, 0], 24, 3024, 72576 [6, 6, 6, 6, 6, 3, 1, 1, 1], 48, 504, 24192 [6, 6, 6, 6, 6, 2, 2, 2, 0], 48, 504, 24192 [6, 6, 6, 6, 6, 2, 2, 1, 1], 96, 756, 72576 [6, 6, 6, 6, 5, 4, 2, 1, 0], 80, 15120, 1209600 [6, 6, 6, 6, 5, 4, 1, 1, 1], 160, 2520, 403200 [6, 6, 6, 6, 5, 3, 3, 1, 0], 160, 7560, 1209600 [6, 6, 6, 6, 5, 3, 2, 2, 0], 320, 7560, 2419200 [6, 6, 6, 6, 5, 3, 2, 1, 1], 640, 7560, 4838400 [6, 6, 6, 6, 5, 2, 2, 2, 1], 1120, 2520, 2822400 [6, 6, 6, 6, 4, 4, 3, 1, 0], 272, 7560, 2056320 [6, 6, 6, 6, 4, 4, 2, 2, 0], 544, 3780, 2056320 [6, 6, 6, 6, 4, 4, 2, 1, 1], 1088, 3780, 4112640 [6, 6, 6, 6, 4, 3, 3, 2, 0], 928, 7560, 7015680 [6, 6, 6, 6, 4, 3, 3, 1, 1], 1856, 3780, 7015680 [6, 6, 6, 6, 4, 3, 2, 2, 1], 3168, 7560, 23950080 [6, 6, 6, 6, 4, 2, 2, 2, 2], 5248, 630, 3306240 [6, 6, 6, 6, 3, 3, 3, 3, 0], 1536, 630, 967680 [6, 6, 6, 6, 3, 3, 3, 2, 1], 5104, 2520, 12862080 [6, 6, 6, 6, 3, 3, 2, 2, 2], 8352, 1260, 10523520 [6, 6, 6, 5, 5, 5, 2, 1, 0], 132, 10080, 1330560 [6, 6, 6, 5, 5, 5, 1, 1, 1], 264, 1680, 443520 [6, 6, 6, 5, 5, 4, 3, 1, 0], 436, 30240, 13184640 [6, 6, 6, 5, 5, 4, 2, 2, 0], 872, 15120, 13184640 [6, 6, 6, 5, 5, 4, 2, 1, 1], 1744, 15120, 26369280 [6, 6, 6, 5, 5, 3, 3, 2, 0], 1480, 15120, 22377600 [6, 6, 6, 5, 5, 3, 3, 1, 1], 2960, 7560, 22377600 [6, 6, 6, 5, 5, 3, 2, 2, 1], 5048, 15120, 76325760 [6, 6, 6, 5, 5, 2, 2, 2, 2], 8352, 1260, 10523520 [6, 6, 6, 5, 4, 4, 4, 1, 0], 712, 10080, 7176960 [6, 6, 6, 5, 4, 4, 3, 2, 0], 2348, 30240, 71003520 [6, 6, 6, 5, 4, 4, 3, 1, 1], 4696, 15120, 71003520 [6, 6, 6, 5, 4, 4, 2, 2, 1], 7968, 15120, 120476160 [6, 6, 6, 5, 4, 3, 3, 3, 0], 3824, 10080, 38545920 [6, 6, 6, 5, 4, 3, 3, 2, 1], 12608, 30240, 381265920 [6, 6, 6, 5, 4, 3, 2, 2, 2], 20520, 10080, 206841600 [6, 6, 6, 5, 3, 3, 3, 3, 1], 19616, 2520, 49432320 [6, 6, 6, 5, 3, 3, 3, 2, 2], 31584, 5040, 159183360 [6, 6, 6, 4, 4, 4, 4, 2, 0], 3664, 2520, 9233280 [6, 6, 6, 4, 4, 4, 4, 1, 1], 7328, 1260, 9233280 [6, 6, 6, 4, 4, 4, 3, 3, 0], 5904, 5040, 29756160 [6, 6, 6, 4, 4, 4, 3, 2, 1], 19324, 10080, 194785920 [6, 6, 6, 4, 4, 4, 2, 2, 2], 31320, 1680, 52617600 [6, 6, 6, 4, 4, 3, 3, 3, 1], 29800, 5040, 150192000 [6, 6, 6, 4, 4, 3, 3, 2, 2], 47792, 7560, 361307520 [6, 6, 6, 4, 3, 3, 3, 3, 2], 72224, 2520, 182004480 [6, 6, 6, 3, 3, 3, 3, 3, 3], 108288, 84, 9096192 [6, 6, 5, 5, 5, 5, 3, 1, 0], 688, 7560, 5201280 [6, 6, 5, 5, 5, 5, 2, 2, 0], 1376, 3780, 5201280 [6, 6, 5, 5, 5, 5, 2, 1, 1], 2752, 3780, 10402560 [6, 6, 5, 5, 5, 4, 4, 1, 0], 1112, 15120, 16813440 [6, 6, 5, 5, 5, 4, 3, 2, 0], 3640, 30240, 110073600 [6, 6, 5, 5, 5, 4, 3, 1, 1], 7280, 15120, 110073600 [6, 6, 5, 5, 5, 4, 2, 2, 1], 12336, 15120, 186520320 [6, 6, 5, 5, 5, 3, 3, 3, 0], 5904, 5040, 29756160 [6, 6, 5, 5, 5, 3, 3, 2, 1], 19432, 15120, 293811840 [6, 6, 5, 5, 5, 3, 2, 2, 2], 31584, 5040, 159183360 [6, 6, 5, 5, 4, 4, 4, 2, 0], 5632, 15120, 85155840 [6, 6, 5, 5, 4, 4, 4, 1, 1], 11264, 7560, 85155840 [6, 6, 5, 5, 4, 4, 3, 3, 0], 9040, 22680, 205027200 [6, 6, 5, 5, 4, 4, 3, 2, 1], 29528, 45360, 1339390080 [6, 6, 5, 5, 4, 4, 2, 2, 2], 47792, 7560, 361307520 [6, 6, 5, 5, 4, 3, 3, 3, 1], 45368, 15120, 685964160 [6, 6, 5, 5, 4, 3, 3, 2, 2], 72656, 22680, 1647838080 [6, 6, 5, 5, 3, 3, 3, 3, 2], 109408, 3780, 413562240 [6, 6, 5, 4, 4, 4, 4, 3, 0], 13712, 7560, 103662720 [6, 6, 5, 4, 4, 4, 4, 2, 1], 44416, 7560, 335784960 [6, 6, 5, 4, 4, 4, 3, 3, 1], 67728, 15120, 1024047360 [6, 6, 5, 4, 4, 4, 3, 2, 2], 108032, 15120, 1633443840 [6, 6, 5, 4, 4, 3, 3, 3, 2], 161704, 15120, 2444964480 [6, 6, 5, 4, 3, 3, 3, 3, 3], 240320, 1512, 363363840 [6, 6, 4, 4, 4, 4, 4, 4, 0], 20640, 252, 5201280 [6, 6, 4, 4, 4, 4, 4, 3, 1], 100368, 1512, 151756416 [6, 6, 4, 4, 4, 4, 4, 2, 2], 159456, 756, 120548736 [6, 6, 4, 4, 4, 4, 3, 3, 2], 237408, 3780, 897402240 [6, 6, 4, 4, 4, 3, 3, 3, 3], 351168, 1260, 442471680 [6, 5, 5, 5, 5, 5, 4, 1, 0], 1720, 3024, 5201280 [6, 5, 5, 5, 5, 5, 3, 2, 0], 5584, 3024, 16886016 [6, 5, 5, 5, 5, 5, 3, 1, 1], 11168, 1512, 16886016 [6, 5, 5, 5, 5, 5, 2, 2, 1], 18896, 1512, 28570752 [6, 5, 5, 5, 5, 4, 4, 2, 0], 8576, 7560, 64834560 [6, 5, 5, 5, 5, 4, 4, 1, 1], 17152, 3780, 64834560 [6, 5, 5, 5, 5, 4, 3, 3, 0], 13712, 7560, 103662720 [6, 5, 5, 5, 5, 4, 3, 2, 1], 44688, 15120, 675682560 [6, 5, 5, 5, 5, 4, 2, 2, 2], 72224, 2520, 182004480 [6, 5, 5, 5, 5, 3, 3, 3, 1], 68416, 2520, 172408320 [6, 5, 5, 5, 5, 3, 3, 2, 2], 109408, 3780, 413562240 [6, 5, 5, 5, 4, 4, 4, 3, 0], 20676, 10080, 208414080 [6, 5, 5, 5, 4, 4, 4, 2, 1], 66804, 10080, 673384320 [6, 5, 5, 5, 4, 4, 3, 3, 1], 101528, 15120, 1535103360 [6, 5, 5, 5, 4, 4, 3, 2, 2], 161704, 15120, 2444964480 [6, 5, 5, 5, 4, 3, 3, 3, 2], 241312, 10080, 2432424960 [6, 5, 5, 5, 3, 3, 3, 3, 3], 357600, 504, 180230400 [6, 5, 5, 4, 4, 4, 4, 4, 0], 30960, 1512, 46811520 [6, 5, 5, 4, 4, 4, 4, 3, 1], 149664, 7560, 1131459840 [6, 5, 5, 4, 4, 4, 4, 2, 2], 237408, 3780, 897402240 [6, 5, 5, 4, 4, 4, 3, 3, 2], 352472, 15120, 5329376640 [6, 5, 5, 4, 4, 3, 3, 3, 3], 519968, 3780, 1965479040 [6, 5, 4, 4, 4, 4, 4, 4, 1], 219360, 504, 110557440 [6, 5, 4, 4, 4, 4, 4, 3, 2], 512184, 3024, 1548844416 [6, 5, 4, 4, 4, 4, 3, 3, 3], 752464, 2520, 1896209280 [6, 4, 4, 4, 4, 4, 4, 4, 2], 740880, 72, 53343360 [6, 4, 4, 4, 4, 4, 4, 3, 3], 1084320, 252, 273248640 [5, 5, 5, 5, 5, 5, 5, 1, 0], 2640, 72, 190080 [5, 5, 5, 5, 5, 5, 4, 2, 0], 12960, 504, 6531840 [5, 5, 5, 5, 5, 5, 4, 1, 1], 25920, 252, 6531840 [5, 5, 5, 5, 5, 5, 3, 3, 0], 20640, 252, 5201280 [5, 5, 5, 5, 5, 5, 3, 2, 1], 67104, 504, 33820416 [5, 5, 5, 5, 5, 5, 2, 2, 2], 108288, 84, 9096192 [5, 5, 5, 5, 5, 4, 4, 3, 0], 30960, 1512, 46811520 [5, 5, 5, 5, 5, 4, 4, 2, 1], 99760, 1512, 150837120 [5, 5, 5, 5, 5, 4, 3, 3, 1], 151120, 1512, 228493440 [5, 5, 5, 5, 5, 4, 3, 2, 2], 240320, 1512, 363363840 [5, 5, 5, 5, 5, 3, 3, 3, 2], 357600, 504, 180230400 [5, 5, 5, 5, 4, 4, 4, 4, 0], 46144, 630, 29070720 [5, 5, 5, 5, 4, 4, 4, 3, 1], 221728, 2520, 558754560 [5, 5, 5, 5, 4, 4, 4, 2, 2], 351168, 1260, 442471680 [5, 5, 5, 5, 4, 4, 3, 3, 2], 519968, 3780, 1965479040 [5, 5, 5, 5, 4, 3, 3, 3, 3], 765120, 630, 482025600 [5, 5, 5, 4, 4, 4, 4, 4, 1], 323600, 504, 163094400 [5, 5, 5, 4, 4, 4, 4, 3, 2], 752464, 2520, 1896209280 [5, 5, 5, 4, 4, 4, 3, 3, 3], 1102824, 1680, 1852744320 [5, 5, 4, 4, 4, 4, 4, 4, 2], 1084320, 252, 273248640 [5, 5, 4, 4, 4, 4, 4, 3, 3], 1583360, 756, 1197020160 [5, 4, 4, 4, 4, 4, 4, 4, 3], 2265200, 72, 163094400 [4, 4, 4, 4, 4, 4, 4, 4, 4], 3230080, 1, 3230080 Sum of numbers in last column=, 68719476736 The most probable vector is, [6, 5, 5, 4, 4, 4, 3, 3, 2], which arises in, 5329376640, out of 2 to the power, 36, that equals, 68719476736, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 9, players The sole champion score-vector is, [6, 5, 5, 4, 4, 4, 3, 3, 2] the corresponding record is, 5329376640 ----------------------------------------------------------------------------\ ----- 10 / 10 \ --------' | --------' | ' | | |' | | | 10, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [9, 8, 7, 6, 5, 4, 3, 2, 1, 0], 1, 3628800, 3628800 [9, 8, 7, 6, 5, 4, 3, 1, 1, 1], 2, 604800, 1209600 [9, 8, 7, 6, 5, 4, 2, 2, 2, 0], 2, 604800, 1209600 [9, 8, 7, 6, 5, 4, 2, 2, 1, 1], 4, 907200, 3628800 [9, 8, 7, 6, 5, 3, 3, 3, 1, 0], 2, 604800, 1209600 [9, 8, 7, 6, 5, 3, 3, 2, 2, 0], 4, 907200, 3628800 [9, 8, 7, 6, 5, 3, 3, 2, 1, 1], 8, 907200, 7257600 [9, 8, 7, 6, 5, 3, 2, 2, 2, 1], 14, 604800, 8467200 [9, 8, 7, 6, 5, 2, 2, 2, 2, 2], 24, 30240, 725760 [9, 8, 7, 6, 4, 4, 4, 2, 1, 0], 2, 604800, 1209600 [9, 8, 7, 6, 4, 4, 4, 1, 1, 1], 4, 100800, 403200 [9, 8, 7, 6, 4, 4, 3, 3, 1, 0], 4, 907200, 3628800 [9, 8, 7, 6, 4, 4, 3, 2, 2, 0], 8, 907200, 7257600 [9, 8, 7, 6, 4, 4, 3, 2, 1, 1], 16, 907200, 14515200 [9, 8, 7, 6, 4, 4, 2, 2, 2, 1], 28, 302400, 8467200 [9, 8, 7, 6, 4, 3, 3, 3, 2, 0], 14, 604800, 8467200 [9, 8, 7, 6, 4, 3, 3, 3, 1, 1], 28, 302400, 8467200 [9, 8, 7, 6, 4, 3, 3, 2, 2, 1], 48, 907200, 43545600 [9, 8, 7, 6, 4, 3, 2, 2, 2, 2], 80, 151200, 12096000 [9, 8, 7, 6, 3, 3, 3, 3, 3, 0], 24, 30240, 725760 [9, 8, 7, 6, 3, 3, 3, 3, 2, 1], 80, 151200, 12096000 [9, 8, 7, 6, 3, 3, 3, 2, 2, 2], 132, 100800, 13305600 [9, 8, 7, 5, 5, 5, 3, 2, 1, 0], 2, 604800, 1209600 [9, 8, 7, 5, 5, 5, 3, 1, 1, 1], 4, 100800, 403200 [9, 8, 7, 5, 5, 5, 2, 2, 2, 0], 4, 100800, 403200 [9, 8, 7, 5, 5, 5, 2, 2, 1, 1], 8, 151200, 1209600 [9, 8, 7, 5, 5, 4, 4, 2, 1, 0], 4, 907200, 3628800 [9, 8, 7, 5, 5, 4, 4, 1, 1, 1], 8, 151200, 1209600 [9, 8, 7, 5, 5, 4, 3, 3, 1, 0], 8, 907200, 7257600 [9, 8, 7, 5, 5, 4, 3, 2, 2, 0], 16, 907200, 14515200 [9, 8, 7, 5, 5, 4, 3, 2, 1, 1], 32, 907200, 29030400 [9, 8, 7, 5, 5, 4, 2, 2, 2, 1], 56, 302400, 16934400 [9, 8, 7, 5, 5, 3, 3, 3, 2, 0], 28, 302400, 8467200 [9, 8, 7, 5, 5, 3, 3, 3, 1, 1], 56, 151200, 8467200 [9, 8, 7, 5, 5, 3, 3, 2, 2, 1], 96, 453600, 43545600 [9, 8, 7, 5, 5, 3, 2, 2, 2, 2], 160, 75600, 12096000 [9, 8, 7, 5, 4, 4, 4, 3, 1, 0], 14, 604800, 8467200 [9, 8, 7, 5, 4, 4, 4, 2, 2, 0], 28, 302400, 8467200 [9, 8, 7, 5, 4, 4, 4, 2, 1, 1], 56, 302400, 16934400 [9, 8, 7, 5, 4, 4, 3, 3, 2, 0], 48, 907200, 43545600 [9, 8, 7, 5, 4, 4, 3, 3, 1, 1], 96, 453600, 43545600 [9, 8, 7, 5, 4, 4, 3, 2, 2, 1], 164, 907200, 148780800 [9, 8, 7, 5, 4, 4, 2, 2, 2, 2], 272, 75600, 20563200 [9, 8, 7, 5, 4, 3, 3, 3, 3, 0], 80, 151200, 12096000 [9, 8, 7, 5, 4, 3, 3, 3, 2, 1], 266, 604800, 160876800 [9, 8, 7, 5, 4, 3, 3, 2, 2, 2], 436, 302400, 131846400 [9, 8, 7, 5, 3, 3, 3, 3, 3, 1], 424, 30240, 12821760 [9, 8, 7, 5, 3, 3, 3, 3, 2, 2], 688, 75600, 52012800 [9, 8, 7, 4, 4, 4, 4, 4, 1, 0], 24, 30240, 725760 [9, 8, 7, 4, 4, 4, 4, 3, 2, 0], 80, 151200, 12096000 [9, 8, 7, 4, 4, 4, 4, 3, 1, 1], 160, 75600, 12096000 [9, 8, 7, 4, 4, 4, 4, 2, 2, 1], 272, 75600, 20563200 [9, 8, 7, 4, 4, 4, 3, 3, 3, 0], 132, 100800, 13305600 [9, 8, 7, 4, 4, 4, 3, 3, 2, 1], 436, 302400, 131846400 [9, 8, 7, 4, 4, 4, 3, 2, 2, 2], 712, 100800, 71769600 [9, 8, 7, 4, 4, 3, 3, 3, 3, 1], 688, 75600, 52012800 [9, 8, 7, 4, 4, 3, 3, 3, 2, 2], 1112, 151200, 168134400 [9, 8, 7, 4, 3, 3, 3, 3, 3, 2], 1720, 30240, 52012800 [9, 8, 7, 3, 3, 3, 3, 3, 3, 3], 2640, 720, 1900800 [9, 8, 6, 6, 6, 4, 3, 2, 1, 0], 2, 604800, 1209600 [9, 8, 6, 6, 6, 4, 3, 1, 1, 1], 4, 100800, 403200 [9, 8, 6, 6, 6, 4, 2, 2, 2, 0], 4, 100800, 403200 [9, 8, 6, 6, 6, 4, 2, 2, 1, 1], 8, 151200, 1209600 [9, 8, 6, 6, 6, 3, 3, 3, 1, 0], 4, 100800, 403200 [9, 8, 6, 6, 6, 3, 3, 2, 2, 0], 8, 151200, 1209600 [9, 8, 6, 6, 6, 3, 3, 2, 1, 1], 16, 151200, 2419200 [9, 8, 6, 6, 6, 3, 2, 2, 2, 1], 28, 100800, 2822400 [9, 8, 6, 6, 6, 2, 2, 2, 2, 2], 48, 5040, 241920 [9, 8, 6, 6, 5, 5, 3, 2, 1, 0], 4, 907200, 3628800 [9, 8, 6, 6, 5, 5, 3, 1, 1, 1], 8, 151200, 1209600 [9, 8, 6, 6, 5, 5, 2, 2, 2, 0], 8, 151200, 1209600 [9, 8, 6, 6, 5, 5, 2, 2, 1, 1], 16, 226800, 3628800 [9, 8, 6, 6, 5, 4, 4, 2, 1, 0], 8, 907200, 7257600 [9, 8, 6, 6, 5, 4, 4, 1, 1, 1], 16, 151200, 2419200 [9, 8, 6, 6, 5, 4, 3, 3, 1, 0], 16, 907200, 14515200 [9, 8, 6, 6, 5, 4, 3, 2, 2, 0], 32, 907200, 29030400 [9, 8, 6, 6, 5, 4, 3, 2, 1, 1], 64, 907200, 58060800 [9, 8, 6, 6, 5, 4, 2, 2, 2, 1], 112, 302400, 33868800 [9, 8, 6, 6, 5, 3, 3, 3, 2, 0], 56, 302400, 16934400 [9, 8, 6, 6, 5, 3, 3, 3, 1, 1], 112, 151200, 16934400 [9, 8, 6, 6, 5, 3, 3, 2, 2, 1], 192, 453600, 87091200 [9, 8, 6, 6, 5, 3, 2, 2, 2, 2], 320, 75600, 24192000 [9, 8, 6, 6, 4, 4, 4, 3, 1, 0], 28, 302400, 8467200 [9, 8, 6, 6, 4, 4, 4, 2, 2, 0], 56, 151200, 8467200 [9, 8, 6, 6, 4, 4, 4, 2, 1, 1], 112, 151200, 16934400 [9, 8, 6, 6, 4, 4, 3, 3, 2, 0], 96, 453600, 43545600 [9, 8, 6, 6, 4, 4, 3, 3, 1, 1], 192, 226800, 43545600 [9, 8, 6, 6, 4, 4, 3, 2, 2, 1], 328, 453600, 148780800 [9, 8, 6, 6, 4, 4, 2, 2, 2, 2], 544, 37800, 20563200 [9, 8, 6, 6, 4, 3, 3, 3, 3, 0], 160, 75600, 12096000 [9, 8, 6, 6, 4, 3, 3, 3, 2, 1], 532, 302400, 160876800 [9, 8, 6, 6, 4, 3, 3, 2, 2, 2], 872, 151200, 131846400 [9, 8, 6, 6, 3, 3, 3, 3, 3, 1], 848, 15120, 12821760 [9, 8, 6, 6, 3, 3, 3, 3, 2, 2], 1376, 37800, 52012800 [9, 8, 6, 5, 5, 5, 4, 2, 1, 0], 14, 604800, 8467200 [9, 8, 6, 5, 5, 5, 4, 1, 1, 1], 28, 100800, 2822400 [9, 8, 6, 5, 5, 5, 3, 3, 1, 0], 28, 302400, 8467200 [9, 8, 6, 5, 5, 5, 3, 2, 2, 0], 56, 302400, 16934400 [9, 8, 6, 5, 5, 5, 3, 2, 1, 1], 112, 302400, 33868800 [9, 8, 6, 5, 5, 5, 2, 2, 2, 1], 196, 100800, 19756800 [9, 8, 6, 5, 5, 4, 4, 3, 1, 0], 48, 907200, 43545600 [9, 8, 6, 5, 5, 4, 4, 2, 2, 0], 96, 453600, 43545600 [9, 8, 6, 5, 5, 4, 4, 2, 1, 1], 192, 453600, 87091200 [9, 8, 6, 5, 5, 4, 3, 3, 2, 0], 164, 907200, 148780800 [9, 8, 6, 5, 5, 4, 3, 3, 1, 1], 328, 453600, 148780800 [9, 8, 6, 5, 5, 4, 3, 2, 2, 1], 560, 907200, 508032000 [9, 8, 6, 5, 5, 4, 2, 2, 2, 2], 928, 75600, 70156800 [9, 8, 6, 5, 5, 3, 3, 3, 3, 0], 272, 75600, 20563200 [9, 8, 6, 5, 5, 3, 3, 3, 2, 1], 904, 302400, 273369600 [9, 8, 6, 5, 5, 3, 3, 2, 2, 2], 1480, 151200, 223776000 [9, 8, 6, 5, 4, 4, 4, 4, 1, 0], 80, 151200, 12096000 [9, 8, 6, 5, 4, 4, 4, 3, 2, 0], 266, 604800, 160876800 [9, 8, 6, 5, 4, 4, 4, 3, 1, 1], 532, 302400, 160876800 [9, 8, 6, 5, 4, 4, 4, 2, 2, 1], 904, 302400, 273369600 [9, 8, 6, 5, 4, 4, 3, 3, 3, 0], 436, 302400, 131846400 [9, 8, 6, 5, 4, 4, 3, 3, 2, 1], 1440, 907200, 1306368000 [9, 8, 6, 5, 4, 4, 3, 2, 2, 2], 2348, 302400, 710035200 [9, 8, 6, 5, 4, 3, 3, 3, 3, 1], 2256, 151200, 341107200 [9, 8, 6, 5, 4, 3, 3, 3, 2, 2], 3640, 302400, 1100736000 [9, 8, 6, 5, 3, 3, 3, 3, 3, 2], 5584, 30240, 168860160 [9, 8, 6, 4, 4, 4, 4, 4, 2, 0], 424, 30240, 12821760 [9, 8, 6, 4, 4, 4, 4, 4, 1, 1], 848, 15120, 12821760 [9, 8, 6, 4, 4, 4, 4, 3, 3, 0], 688, 75600, 52012800 [9, 8, 6, 4, 4, 4, 4, 3, 2, 1], 2256, 151200, 341107200 [9, 8, 6, 4, 4, 4, 4, 2, 2, 2], 3664, 25200, 92332800 [9, 8, 6, 4, 4, 4, 3, 3, 3, 1], 3504, 100800, 353203200 [9, 8, 6, 4, 4, 4, 3, 3, 2, 2], 5632, 151200, 851558400 [9, 8, 6, 4, 4, 3, 3, 3, 3, 2], 8576, 75600, 648345600 [9, 8, 6, 4, 3, 3, 3, 3, 3, 3], 12960, 5040, 65318400 [9, 8, 5, 5, 5, 5, 5, 2, 1, 0], 24, 30240, 725760 [9, 8, 5, 5, 5, 5, 5, 1, 1, 1], 48, 5040, 241920 [9, 8, 5, 5, 5, 5, 4, 3, 1, 0], 80, 151200, 12096000 [9, 8, 5, 5, 5, 5, 4, 2, 2, 0], 160, 75600, 12096000 [9, 8, 5, 5, 5, 5, 4, 2, 1, 1], 320, 75600, 24192000 [9, 8, 5, 5, 5, 5, 3, 3, 2, 0], 272, 75600, 20563200 [9, 8, 5, 5, 5, 5, 3, 3, 1, 1], 544, 37800, 20563200 [9, 8, 5, 5, 5, 5, 3, 2, 2, 1], 928, 75600, 70156800 [9, 8, 5, 5, 5, 5, 2, 2, 2, 2], 1536, 6300, 9676800 [9, 8, 5, 5, 5, 4, 4, 4, 1, 0], 132, 100800, 13305600 [9, 8, 5, 5, 5, 4, 4, 3, 2, 0], 436, 302400, 131846400 [9, 8, 5, 5, 5, 4, 4, 3, 1, 1], 872, 151200, 131846400 [9, 8, 5, 5, 5, 4, 4, 2, 2, 1], 1480, 151200, 223776000 [9, 8, 5, 5, 5, 4, 3, 3, 3, 0], 712, 100800, 71769600 [9, 8, 5, 5, 5, 4, 3, 3, 2, 1], 2348, 302400, 710035200 [9, 8, 5, 5, 5, 4, 3, 2, 2, 2], 3824, 100800, 385459200 [9, 8, 5, 5, 5, 3, 3, 3, 3, 1], 3664, 25200, 92332800 [9, 8, 5, 5, 5, 3, 3, 3, 2, 2], 5904, 50400, 297561600 [9, 8, 5, 5, 4, 4, 4, 4, 2, 0], 688, 75600, 52012800 [9, 8, 5, 5, 4, 4, 4, 4, 1, 1], 1376, 37800, 52012800 [9, 8, 5, 5, 4, 4, 4, 3, 3, 0], 1112, 151200, 168134400 [9, 8, 5, 5, 4, 4, 4, 3, 2, 1], 3640, 302400, 1100736000 [9, 8, 5, 5, 4, 4, 4, 2, 2, 2], 5904, 50400, 297561600 [9, 8, 5, 5, 4, 4, 3, 3, 3, 1], 5632, 151200, 851558400 [9, 8, 5, 5, 4, 4, 3, 3, 2, 2], 9040, 226800, 2050272000 [9, 8, 5, 5, 4, 3, 3, 3, 3, 2], 13712, 75600, 1036627200 [9, 8, 5, 5, 3, 3, 3, 3, 3, 3], 20640, 2520, 52012800 [9, 8, 5, 4, 4, 4, 4, 4, 3, 0], 1720, 30240, 52012800 [9, 8, 5, 4, 4, 4, 4, 4, 2, 1], 5584, 30240, 168860160 [9, 8, 5, 4, 4, 4, 4, 3, 3, 1], 8576, 75600, 648345600 [9, 8, 5, 4, 4, 4, 4, 3, 2, 2], 13712, 75600, 1036627200 [9, 8, 5, 4, 4, 4, 3, 3, 3, 2], 20676, 100800, 2084140800 [9, 8, 5, 4, 4, 3, 3, 3, 3, 3], 30960, 15120, 468115200 [9, 8, 4, 4, 4, 4, 4, 4, 4, 0], 2640, 720, 1900800 [9, 8, 4, 4, 4, 4, 4, 4, 3, 1], 12960, 5040, 65318400 [9, 8, 4, 4, 4, 4, 4, 4, 2, 2], 20640, 2520, 52012800 [9, 8, 4, 4, 4, 4, 4, 3, 3, 2], 30960, 15120, 468115200 [9, 8, 4, 4, 4, 4, 3, 3, 3, 3], 46144, 6300, 290707200 [9, 7, 7, 7, 5, 4, 3, 2, 1, 0], 2, 604800, 1209600 [9, 7, 7, 7, 5, 4, 3, 1, 1, 1], 4, 100800, 403200 [9, 7, 7, 7, 5, 4, 2, 2, 2, 0], 4, 100800, 403200 [9, 7, 7, 7, 5, 4, 2, 2, 1, 1], 8, 151200, 1209600 [9, 7, 7, 7, 5, 3, 3, 3, 1, 0], 4, 100800, 403200 [9, 7, 7, 7, 5, 3, 3, 2, 2, 0], 8, 151200, 1209600 [9, 7, 7, 7, 5, 3, 3, 2, 1, 1], 16, 151200, 2419200 [9, 7, 7, 7, 5, 3, 2, 2, 2, 1], 28, 100800, 2822400 [9, 7, 7, 7, 5, 2, 2, 2, 2, 2], 48, 5040, 241920 [9, 7, 7, 7, 4, 4, 4, 2, 1, 0], 4, 100800, 403200 [9, 7, 7, 7, 4, 4, 4, 1, 1, 1], 8, 16800, 134400 [9, 7, 7, 7, 4, 4, 3, 3, 1, 0], 8, 151200, 1209600 [9, 7, 7, 7, 4, 4, 3, 2, 2, 0], 16, 151200, 2419200 [9, 7, 7, 7, 4, 4, 3, 2, 1, 1], 32, 151200, 4838400 [9, 7, 7, 7, 4, 4, 2, 2, 2, 1], 56, 50400, 2822400 [9, 7, 7, 7, 4, 3, 3, 3, 2, 0], 28, 100800, 2822400 [9, 7, 7, 7, 4, 3, 3, 3, 1, 1], 56, 50400, 2822400 [9, 7, 7, 7, 4, 3, 3, 2, 2, 1], 96, 151200, 14515200 [9, 7, 7, 7, 4, 3, 2, 2, 2, 2], 160, 25200, 4032000 [9, 7, 7, 7, 3, 3, 3, 3, 3, 0], 48, 5040, 241920 [9, 7, 7, 7, 3, 3, 3, 3, 2, 1], 160, 25200, 4032000 [9, 7, 7, 7, 3, 3, 3, 2, 2, 2], 264, 16800, 4435200 [9, 7, 7, 6, 6, 4, 3, 2, 1, 0], 4, 907200, 3628800 [9, 7, 7, 6, 6, 4, 3, 1, 1, 1], 8, 151200, 1209600 [9, 7, 7, 6, 6, 4, 2, 2, 2, 0], 8, 151200, 1209600 [9, 7, 7, 6, 6, 4, 2, 2, 1, 1], 16, 226800, 3628800 [9, 7, 7, 6, 6, 3, 3, 3, 1, 0], 8, 151200, 1209600 [9, 7, 7, 6, 6, 3, 3, 2, 2, 0], 16, 226800, 3628800 [9, 7, 7, 6, 6, 3, 3, 2, 1, 1], 32, 226800, 7257600 [9, 7, 7, 6, 6, 3, 2, 2, 2, 1], 56, 151200, 8467200 [9, 7, 7, 6, 6, 2, 2, 2, 2, 2], 96, 7560, 725760 [9, 7, 7, 6, 5, 5, 3, 2, 1, 0], 8, 907200, 7257600 [9, 7, 7, 6, 5, 5, 3, 1, 1, 1], 16, 151200, 2419200 [9, 7, 7, 6, 5, 5, 2, 2, 2, 0], 16, 151200, 2419200 [9, 7, 7, 6, 5, 5, 2, 2, 1, 1], 32, 226800, 7257600 [9, 7, 7, 6, 5, 4, 4, 2, 1, 0], 16, 907200, 14515200 [9, 7, 7, 6, 5, 4, 4, 1, 1, 1], 32, 151200, 4838400 [9, 7, 7, 6, 5, 4, 3, 3, 1, 0], 32, 907200, 29030400 [9, 7, 7, 6, 5, 4, 3, 2, 2, 0], 64, 907200, 58060800 [9, 7, 7, 6, 5, 4, 3, 2, 1, 1], 128, 907200, 116121600 [9, 7, 7, 6, 5, 4, 2, 2, 2, 1], 224, 302400, 67737600 [9, 7, 7, 6, 5, 3, 3, 3, 2, 0], 112, 302400, 33868800 [9, 7, 7, 6, 5, 3, 3, 3, 1, 1], 224, 151200, 33868800 [9, 7, 7, 6, 5, 3, 3, 2, 2, 1], 384, 453600, 174182400 [9, 7, 7, 6, 5, 3, 2, 2, 2, 2], 640, 75600, 48384000 [9, 7, 7, 6, 4, 4, 4, 3, 1, 0], 56, 302400, 16934400 [9, 7, 7, 6, 4, 4, 4, 2, 2, 0], 112, 151200, 16934400 [9, 7, 7, 6, 4, 4, 4, 2, 1, 1], 224, 151200, 33868800 [9, 7, 7, 6, 4, 4, 3, 3, 2, 0], 192, 453600, 87091200 [9, 7, 7, 6, 4, 4, 3, 3, 1, 1], 384, 226800, 87091200 [9, 7, 7, 6, 4, 4, 3, 2, 2, 1], 656, 453600, 297561600 [9, 7, 7, 6, 4, 4, 2, 2, 2, 2], 1088, 37800, 41126400 [9, 7, 7, 6, 4, 3, 3, 3, 3, 0], 320, 75600, 24192000 [9, 7, 7, 6, 4, 3, 3, 3, 2, 1], 1064, 302400, 321753600 [9, 7, 7, 6, 4, 3, 3, 2, 2, 2], 1744, 151200, 263692800 [9, 7, 7, 6, 3, 3, 3, 3, 3, 1], 1696, 15120, 25643520 [9, 7, 7, 6, 3, 3, 3, 3, 2, 2], 2752, 37800, 104025600 [9, 7, 7, 5, 5, 5, 4, 2, 1, 0], 28, 302400, 8467200 [9, 7, 7, 5, 5, 5, 4, 1, 1, 1], 56, 50400, 2822400 [9, 7, 7, 5, 5, 5, 3, 3, 1, 0], 56, 151200, 8467200 [9, 7, 7, 5, 5, 5, 3, 2, 2, 0], 112, 151200, 16934400 [9, 7, 7, 5, 5, 5, 3, 2, 1, 1], 224, 151200, 33868800 [9, 7, 7, 5, 5, 5, 2, 2, 2, 1], 392, 50400, 19756800 [9, 7, 7, 5, 5, 4, 4, 3, 1, 0], 96, 453600, 43545600 [9, 7, 7, 5, 5, 4, 4, 2, 2, 0], 192, 226800, 43545600 [9, 7, 7, 5, 5, 4, 4, 2, 1, 1], 384, 226800, 87091200 [9, 7, 7, 5, 5, 4, 3, 3, 2, 0], 328, 453600, 148780800 [9, 7, 7, 5, 5, 4, 3, 3, 1, 1], 656, 226800, 148780800 [9, 7, 7, 5, 5, 4, 3, 2, 2, 1], 1120, 453600, 508032000 [9, 7, 7, 5, 5, 4, 2, 2, 2, 2], 1856, 37800, 70156800 [9, 7, 7, 5, 5, 3, 3, 3, 3, 0], 544, 37800, 20563200 [9, 7, 7, 5, 5, 3, 3, 3, 2, 1], 1808, 151200, 273369600 [9, 7, 7, 5, 5, 3, 3, 2, 2, 2], 2960, 75600, 223776000 [9, 7, 7, 5, 4, 4, 4, 4, 1, 0], 160, 75600, 12096000 [9, 7, 7, 5, 4, 4, 4, 3, 2, 0], 532, 302400, 160876800 [9, 7, 7, 5, 4, 4, 4, 3, 1, 1], 1064, 151200, 160876800 [9, 7, 7, 5, 4, 4, 4, 2, 2, 1], 1808, 151200, 273369600 [9, 7, 7, 5, 4, 4, 3, 3, 3, 0], 872, 151200, 131846400 [9, 7, 7, 5, 4, 4, 3, 3, 2, 1], 2880, 453600, 1306368000 [9, 7, 7, 5, 4, 4, 3, 2, 2, 2], 4696, 151200, 710035200 [9, 7, 7, 5, 4, 3, 3, 3, 3, 1], 4512, 75600, 341107200 [9, 7, 7, 5, 4, 3, 3, 3, 2, 2], 7280, 151200, 1100736000 [9, 7, 7, 5, 3, 3, 3, 3, 3, 2], 11168, 15120, 168860160 [9, 7, 7, 4, 4, 4, 4, 4, 2, 0], 848, 15120, 12821760 [9, 7, 7, 4, 4, 4, 4, 4, 1, 1], 1696, 7560, 12821760 [9, 7, 7, 4, 4, 4, 4, 3, 3, 0], 1376, 37800, 52012800 [9, 7, 7, 4, 4, 4, 4, 3, 2, 1], 4512, 75600, 341107200 [9, 7, 7, 4, 4, 4, 4, 2, 2, 2], 7328, 12600, 92332800 [9, 7, 7, 4, 4, 4, 3, 3, 3, 1], 7008, 50400, 353203200 [9, 7, 7, 4, 4, 4, 3, 3, 2, 2], 11264, 75600, 851558400 [9, 7, 7, 4, 4, 3, 3, 3, 3, 2], 17152, 37800, 648345600 [9, 7, 7, 4, 3, 3, 3, 3, 3, 3], 25920, 2520, 65318400 [9, 7, 6, 6, 6, 5, 3, 2, 1, 0], 14, 604800, 8467200 [9, 7, 6, 6, 6, 5, 3, 1, 1, 1], 28, 100800, 2822400 [9, 7, 6, 6, 6, 5, 2, 2, 2, 0], 28, 100800, 2822400 [9, 7, 6, 6, 6, 5, 2, 2, 1, 1], 56, 151200, 8467200 [9, 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5040, 20723673600 [7, 6, 6, 6, 4, 4, 4, 4, 4, 0], 357600, 5040, 1802304000 [7, 6, 6, 6, 4, 4, 4, 4, 3, 1], 1719488, 25200, 43331097600 [7, 6, 6, 6, 4, 4, 4, 4, 2, 2], 2723776, 12600, 34319577600 [7, 6, 6, 6, 4, 4, 4, 3, 3, 2], 4026920, 50400, 202956768000 [7, 6, 6, 6, 4, 4, 3, 3, 3, 3], 5915104, 12600, 74530310400 [7, 6, 6, 5, 5, 5, 5, 5, 1, 0], 30960, 15120, 468115200 [7, 6, 6, 5, 5, 5, 5, 4, 2, 0], 149664, 75600, 11314598400 [7, 6, 6, 5, 5, 5, 5, 4, 1, 1], 299328, 37800, 11314598400 [7, 6, 6, 5, 5, 5, 5, 3, 3, 0], 237408, 37800, 8974022400 [7, 6, 6, 5, 5, 5, 5, 3, 2, 1], 770912, 75600, 58280947200 [7, 6, 6, 5, 5, 5, 5, 2, 2, 2], 1242496, 12600, 15655449600 [7, 6, 6, 5, 5, 5, 4, 4, 3, 0], 352472, 151200, 53293766400 [7, 6, 6, 5, 5, 5, 4, 4, 2, 1], 1134280, 151200, 171503136000 [7, 6, 6, 5, 5, 5, 4, 3, 3, 1], 1711256, 151200, 258741907200 [7, 6, 6, 5, 5, 5, 4, 3, 2, 2], 2717568, 151200, 410896281600 [7, 6, 6, 5, 5, 5, 3, 3, 3, 2], 4026920, 50400, 202956768000 [7, 6, 6, 5, 5, 4, 4, 4, 4, 0], 519968, 37800, 19654790400 [7, 6, 6, 5, 5, 4, 4, 4, 3, 1], 2484944, 151200, 375723532800 [7, 6, 6, 5, 5, 4, 4, 4, 2, 2], 3929952, 75600, 297104371200 [7, 6, 6, 5, 5, 4, 4, 3, 3, 2], 5794400, 226800, 1314169920000 [7, 6, 6, 5, 5, 4, 3, 3, 3, 3], 8489696, 37800, 320910508800 [7, 6, 6, 5, 4, 4, 4, 4, 4, 1], 3589200, 15120, 54268704000 [7, 6, 6, 5, 4, 4, 4, 4, 3, 2], 8297888, 75600, 627320332800 [7, 6, 6, 5, 4, 4, 4, 3, 3, 3], 12108512, 50400, 610269004800 [7, 6, 6, 4, 4, 4, 4, 4, 4, 2], 11832576, 2520, 29818091520 [7, 6, 6, 4, 4, 4, 4, 4, 3, 3], 17201952, 7560, 130046757120 [7, 6, 5, 5, 5, 5, 5, 5, 2, 0], 219360, 5040, 1105574400 [7, 6, 5, 5, 5, 5, 5, 5, 1, 1], 438720, 2520, 1105574400 [7, 6, 5, 5, 5, 5, 5, 4, 3, 0], 512184, 30240, 15488444160 [7, 6, 5, 5, 5, 5, 5, 4, 2, 1], 1643376, 30240, 49695690240 [7, 6, 5, 5, 5, 5, 5, 3, 3, 1], 2471232, 15120, 37365027840 [7, 6, 5, 5, 5, 5, 5, 3, 2, 2], 3918096, 15120, 59241611520 [7, 6, 5, 5, 5, 5, 4, 4, 4, 0], 752464, 25200, 18962092800 [7, 6, 5, 5, 5, 5, 4, 4, 3, 1], 3573792, 75600, 270178675200 [7, 6, 5, 5, 5, 5, 4, 4, 2, 2], 5642656, 37800, 213292396800 [7, 6, 5, 5, 5, 5, 4, 3, 3, 2], 8297888, 75600, 627320332800 [7, 6, 5, 5, 5, 5, 3, 3, 3, 3], 12128128, 6300, 76407206400 [7, 6, 5, 5, 5, 4, 4, 4, 4, 1], 5142656, 25200, 129594931200 [7, 6, 5, 5, 5, 4, 4, 4, 3, 2], 11840644, 100800, 1193536915200 [7, 6, 5, 5, 5, 4, 4, 3, 3, 3], 17238648, 50400, 868827859200 [7, 6, 5, 5, 4, 4, 4, 4, 4, 2], 16829040, 15120, 254455084800 [7, 6, 5, 5, 4, 4, 4, 4, 3, 3], 24412704, 37800, 922800211200 [7, 6, 5, 4, 4, 4, 4, 4, 4, 3], 34456544, 5040, 173660981760 [7, 6, 4, 4, 4, 4, 4, 4, 4, 4], 48482560, 90, 4363430400 [7, 5, 5, 5, 5, 5, 5, 5, 3, 0], 740880, 720, 533433600 [7, 5, 5, 5, 5, 5, 5, 5, 2, 1], 2369760, 720, 1706227200 [7, 5, 5, 5, 5, 5, 5, 4, 4, 0], 1084320, 2520, 2732486400 [7, 5, 5, 5, 5, 5, 5, 4, 3, 1], 5117664, 5040, 25793026560 [7, 5, 5, 5, 5, 5, 5, 4, 2, 2], 8066688, 2520, 20328053760 [7, 5, 5, 5, 5, 5, 5, 3, 3, 2], 11832576, 2520, 29818091520 [7, 5, 5, 5, 5, 5, 4, 4, 4, 1], 7338880, 5040, 36987955200 [7, 5, 5, 5, 5, 5, 4, 4, 3, 2], 16829040, 15120, 254455084800 [7, 5, 5, 5, 5, 5, 4, 3, 3, 3], 24447120, 5040, 123213484800 [7, 5, 5, 5, 5, 4, 4, 4, 4, 2], 23846464, 6300, 150232723200 [7, 5, 5, 5, 5, 4, 4, 4, 3, 3], 34520096, 12600, 434953209600 [7, 5, 5, 5, 4, 4, 4, 4, 4, 3], 48589520, 5040, 244891180800 [7, 5, 5, 4, 4, 4, 4, 4, 4, 4], 68193440, 360, 24549638400 [6, 6, 6, 6, 6, 6, 6, 2, 1, 0], 2640, 720, 1900800 [6, 6, 6, 6, 6, 6, 6, 1, 1, 1], 5280, 120, 633600 [6, 6, 6, 6, 6, 6, 5, 3, 1, 0], 12960, 5040, 65318400 [6, 6, 6, 6, 6, 6, 5, 2, 2, 0], 25920, 2520, 65318400 [6, 6, 6, 6, 6, 6, 5, 2, 1, 1], 51840, 2520, 130636800 [6, 6, 6, 6, 6, 6, 4, 4, 1, 0], 20640, 2520, 52012800 [6, 6, 6, 6, 6, 6, 4, 3, 2, 0], 67104, 5040, 338204160 [6, 6, 6, 6, 6, 6, 4, 3, 1, 1], 134208, 2520, 338204160 [6, 6, 6, 6, 6, 6, 4, 2, 2, 1], 227136, 2520, 572382720 [6, 6, 6, 6, 6, 6, 3, 3, 3, 0], 108288, 840, 90961920 [6, 6, 6, 6, 6, 6, 3, 3, 2, 1], 355872, 2520, 896797440 [6, 6, 6, 6, 6, 6, 3, 2, 2, 2], 577536, 840, 485130240 [6, 6, 6, 6, 6, 5, 5, 4, 1, 0], 30960, 15120, 468115200 [6, 6, 6, 6, 6, 5, 5, 3, 2, 0], 99760, 15120, 1508371200 [6, 6, 6, 6, 6, 5, 5, 3, 1, 1], 199520, 7560, 1508371200 [6, 6, 6, 6, 6, 5, 5, 2, 2, 1], 337120, 7560, 2548627200 [6, 6, 6, 6, 6, 5, 4, 4, 2, 0], 151120, 15120, 2284934400 [6, 6, 6, 6, 6, 5, 4, 4, 1, 1], 302240, 7560, 2284934400 [6, 6, 6, 6, 6, 5, 4, 3, 3, 0], 240320, 15120, 3633638400 [6, 6, 6, 6, 6, 5, 4, 3, 2, 1], 781744, 30240, 23639938560 [6, 6, 6, 6, 6, 5, 4, 2, 2, 2], 1261248, 5040, 6356689920 [6, 6, 6, 6, 6, 5, 3, 3, 3, 1], 1190368, 5040, 5999454720 [6, 6, 6, 6, 6, 5, 3, 3, 2, 2], 1900096, 7560, 14364725760 [6, 6, 6, 6, 6, 4, 4, 4, 3, 0], 357600, 5040, 1802304000 [6, 6, 6, 6, 6, 4, 4, 4, 2, 1], 1153040, 5040, 5811321600 [6, 6, 6, 6, 6, 4, 4, 3, 3, 1], 1742880, 7560, 13176172800 [6, 6, 6, 6, 6, 4, 4, 3, 2, 2], 2770560, 7560, 20945433600 [6, 6, 6, 6, 6, 4, 3, 3, 3, 2], 4111840, 5040, 20723673600 [6, 6, 6, 6, 6, 3, 3, 3, 3, 3], 6059520, 252, 1526999040 [6, 6, 6, 6, 5, 5, 5, 5, 1, 0], 46144, 6300, 290707200 [6, 6, 6, 6, 5, 5, 5, 4, 2, 0], 221728, 25200, 5587545600 [6, 6, 6, 6, 5, 5, 5, 4, 1, 1], 443456, 12600, 5587545600 [6, 6, 6, 6, 5, 5, 5, 3, 3, 0], 351168, 12600, 4424716800 [6, 6, 6, 6, 5, 5, 5, 3, 2, 1], 1139392, 25200, 28712678400 [6, 6, 6, 6, 5, 5, 5, 2, 2, 2], 1835328, 4200, 7708377600 [6, 6, 6, 6, 5, 5, 4, 4, 3, 0], 519968, 37800, 19654790400 [6, 6, 6, 6, 5, 5, 4, 4, 2, 1], 1671776, 37800, 63193132800 [6, 6, 6, 6, 5, 5, 4, 3, 3, 1], 2518560, 37800, 95201568000 [6, 6, 6, 6, 5, 5, 4, 3, 2, 2], 3997184, 37800, 151093555200 [6, 6, 6, 6, 5, 5, 3, 3, 3, 2], 5915104, 12600, 74530310400 [6, 6, 6, 6, 5, 4, 4, 4, 4, 0], 765120, 6300, 4820256000 [6, 6, 6, 6, 5, 4, 4, 4, 3, 1], 3647872, 25200, 91926374400 [6, 6, 6, 6, 5, 4, 4, 4, 2, 2], 5765504, 12600, 72645350400 [6, 6, 6, 6, 5, 4, 4, 3, 3, 2], 8489696, 37800, 320910508800 [6, 6, 6, 6, 5, 4, 3, 3, 3, 3], 12422912, 6300, 78264345600 [6, 6, 6, 6, 4, 4, 4, 4, 4, 1], 5256000, 1260, 6622560000 [6, 6, 6, 6, 4, 4, 4, 4, 3, 2], 12128128, 6300, 76407206400 [6, 6, 6, 6, 4, 4, 4, 3, 3, 3], 17675712, 4200, 74237990400 [6, 6, 6, 5, 5, 5, 5, 5, 2, 0], 323600, 5040, 1630944000 [6, 6, 6, 5, 5, 5, 5, 5, 1, 1], 647200, 2520, 1630944000 [6, 6, 6, 5, 5, 5, 5, 4, 3, 0], 752464, 25200, 18962092800 [6, 6, 6, 5, 5, 5, 5, 4, 2, 1], 2411968, 25200, 60781593600 [6, 6, 6, 5, 5, 5, 5, 3, 3, 1], 3621824, 12600, 45634982400 [6, 6, 6, 5, 5, 5, 5, 3, 2, 2], 5738720, 12600, 72307872000 [6, 6, 6, 5, 5, 5, 4, 4, 4, 0], 1102824, 16800, 18527443200 [6, 6, 6, 5, 5, 5, 4, 4, 3, 1], 5225072, 50400, 263343628800 [6, 6, 6, 5, 5, 5, 4, 4, 2, 2], 8244496, 25200, 207761299200 [6, 6, 6, 5, 5, 5, 4, 3, 3, 2], 12108512, 50400, 610269004800 [6, 6, 6, 5, 5, 5, 3, 3, 3, 3], 17675712, 4200, 74237990400 [6, 6, 6, 5, 5, 4, 4, 4, 4, 1], 7501440, 12600, 94518144000 [6, 6, 6, 5, 5, 4, 4, 4, 3, 2], 17238648, 50400, 868827859200 [6, 6, 6, 5, 5, 4, 4, 3, 3, 3], 25067088, 25200, 631690617600 [6, 6, 6, 5, 4, 4, 4, 4, 4, 2], 24447120, 5040, 123213484800 [6, 6, 6, 5, 4, 4, 4, 4, 3, 3], 35421600, 12600, 446312160000 [6, 6, 6, 4, 4, 4, 4, 4, 4, 3], 49889088, 840, 41906833920 [6, 6, 5, 5, 5, 5, 5, 5, 3, 0], 1084320, 2520, 2732486400 [6, 6, 5, 5, 5, 5, 5, 5, 2, 1], 3464640, 2520, 8730892800 [6, 6, 5, 5, 5, 5, 5, 4, 4, 0], 1583360, 7560, 11970201600 [6, 6, 5, 5, 5, 5, 5, 4, 3, 1], 7454368, 15120, 112710044160 [6, 6, 5, 5, 5, 5, 5, 4, 2, 2], 11742016, 7560, 88769640960 [6, 6, 5, 5, 5, 5, 5, 3, 3, 2], 17201952, 7560, 130046757120 [6, 6, 5, 5, 5, 5, 4, 4, 4, 1], 10666304, 12600, 134395430400 [6, 6, 5, 5, 5, 5, 4, 4, 3, 2], 24412704, 37800, 922800211200 [6, 6, 5, 5, 5, 5, 4, 3, 3, 3], 35421600, 12600, 446312160000 [6, 6, 5, 5, 5, 4, 4, 4, 4, 2], 34520096, 12600, 434953209600 [6, 6, 5, 5, 5, 4, 4, 4, 3, 3], 49913232, 25200, 1257813446400 [6, 6, 5, 5, 4, 4, 4, 4, 4, 3], 70116000, 7560, 530076960000 [6, 6, 5, 4, 4, 4, 4, 4, 4, 4], 98213920, 360, 35357011200 [6, 5, 5, 5, 5, 5, 5, 5, 4, 0], 2265200, 720, 1630944000 [6, 5, 5, 5, 5, 5, 5, 5, 3, 1], 10596320, 720, 7629350400 [6, 5, 5, 5, 5, 5, 5, 5, 2, 2], 16662240, 360, 5998406400 [6, 5, 5, 5, 5, 5, 5, 4, 4, 1], 15114880, 2520, 38089497600 [6, 5, 5, 5, 5, 5, 5, 4, 3, 2], 34456544, 5040, 173660981760 [6, 5, 5, 5, 5, 5, 5, 3, 3, 3], 49889088, 840, 41906833920 [6, 5, 5, 5, 5, 5, 4, 4, 4, 2], 48589520, 5040, 244891180800 [6, 5, 5, 5, 5, 5, 4, 4, 3, 3], 70116000, 7560, 530076960000 [6, 5, 5, 5, 5, 4, 4, 4, 4, 3], 98254848, 6300, 619005542400 [6, 5, 5, 5, 4, 4, 4, 4, 4, 4], 137311360, 840, 115341542400 [5, 5, 5, 5, 5, 5, 5, 5, 5, 0], 3230080, 10, 32300800 [5, 5, 5, 5, 5, 5, 5, 5, 4, 1], 21351680, 90, 1921651200 [5, 5, 5, 5, 5, 5, 5, 5, 3, 2], 48482560, 90, 4363430400 [5, 5, 5, 5, 5, 5, 5, 4, 4, 2], 68193440, 360, 24549638400 [5, 5, 5, 5, 5, 5, 5, 4, 3, 3], 98213920, 360, 35357011200 [5, 5, 5, 5, 5, 5, 4, 4, 4, 3], 137311360, 840, 115341542400 [5, 5, 5, 5, 5, 4, 4, 4, 4, 4], 191474240, 252, 48251508480 Sum of numbers in last column=, 35184372088832 The most probable vector is, [7, 6, 6, 5, 5, 4, 4, 3, 3, 2], which arises in, 1314169920000, out of 2 to the power, 45, that equals, 35184372088832, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 10, players The sole champion score-vector is, [7, 6, 6, 5, 5, 4, 4, 3, 3, 2] the corresponding record is, 1314169920000 ----------------------------------------------------------------------------\ ----- 11 / 11 \ --------' | --------' | ' | | |' | | | 11, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], 1, 39916800, 39916800 [10, 9, 8, 7, 6, 5, 4, 3, 1, 1, 1], 2, 6652800, 13305600 [10, 9, 8, 7, 6, 5, 4, 2, 2, 2, 0], 2, 6652800, 13305600 [10, 9, 8, 7, 6, 5, 4, 2, 2, 1, 1], 4, 9979200, 39916800 [10, 9, 8, 7, 6, 5, 3, 3, 3, 1, 0], 2, 6652800, 13305600 [10, 9, 8, 7, 6, 5, 3, 3, 2, 2, 0], 4, 9979200, 39916800 [10, 9, 8, 7, 6, 5, 3, 3, 2, 1, 1], 8, 9979200, 79833600 [10, 9, 8, 7, 6, 5, 3, 2, 2, 2, 1], 14, 6652800, 93139200 [10, 9, 8, 7, 6, 5, 2, 2, 2, 2, 2], 24, 332640, 7983360 [10, 9, 8, 7, 6, 4, 4, 4, 2, 1, 0], 2, 6652800, 13305600 [10, 9, 8, 7, 6, 4, 4, 4, 1, 1, 1], 4, 1108800, 4435200 [10, 9, 8, 7, 6, 4, 4, 3, 3, 1, 0], 4, 9979200, 39916800 [10, 9, 8, 7, 6, 4, 4, 3, 2, 2, 0], 8, 9979200, 79833600 [10, 9, 8, 7, 6, 4, 4, 3, 2, 1, 1], 16, 9979200, 159667200 [10, 9, 8, 7, 6, 4, 4, 2, 2, 2, 1], 28, 3326400, 93139200 [10, 9, 8, 7, 6, 4, 3, 3, 3, 2, 0], 14, 6652800, 93139200 [10, 9, 8, 7, 6, 4, 3, 3, 3, 1, 1], 28, 3326400, 93139200 [10, 9, 8, 7, 6, 4, 3, 3, 2, 2, 1], 48, 9979200, 479001600 [10, 9, 8, 7, 6, 4, 3, 2, 2, 2, 2], 80, 1663200, 133056000 [10, 9, 8, 7, 6, 3, 3, 3, 3, 3, 0], 24, 332640, 7983360 [10, 9, 8, 7, 6, 3, 3, 3, 3, 2, 1], 80, 1663200, 133056000 [10, 9, 8, 7, 6, 3, 3, 3, 2, 2, 2], 132, 1108800, 146361600 [10, 9, 8, 7, 5, 5, 5, 3, 2, 1, 0], 2, 6652800, 13305600 [10, 9, 8, 7, 5, 5, 5, 3, 1, 1, 1], 4, 1108800, 4435200 [10, 9, 8, 7, 5, 5, 5, 2, 2, 2, 0], 4, 1108800, 4435200 [10, 9, 8, 7, 5, 5, 5, 2, 2, 1, 1], 8, 1663200, 13305600 [10, 9, 8, 7, 5, 5, 4, 4, 2, 1, 0], 4, 9979200, 39916800 [10, 9, 8, 7, 5, 5, 4, 4, 1, 1, 1], 8, 1663200, 13305600 [10, 9, 8, 7, 5, 5, 4, 3, 3, 1, 0], 8, 9979200, 79833600 [10, 9, 8, 7, 5, 5, 4, 3, 2, 2, 0], 16, 9979200, 159667200 [10, 9, 8, 7, 5, 5, 4, 3, 2, 1, 1], 32, 9979200, 319334400 [10, 9, 8, 7, 5, 5, 4, 2, 2, 2, 1], 56, 3326400, 186278400 [10, 9, 8, 7, 5, 5, 3, 3, 3, 2, 0], 28, 3326400, 93139200 [10, 9, 8, 7, 5, 5, 3, 3, 3, 1, 1], 56, 1663200, 93139200 [10, 9, 8, 7, 5, 5, 3, 3, 2, 2, 1], 96, 4989600, 479001600 [10, 9, 8, 7, 5, 5, 3, 2, 2, 2, 2], 160, 831600, 133056000 [10, 9, 8, 7, 5, 4, 4, 4, 3, 1, 0], 14, 6652800, 93139200 [10, 9, 8, 7, 5, 4, 4, 4, 2, 2, 0], 28, 3326400, 93139200 [10, 9, 8, 7, 5, 4, 4, 4, 2, 1, 1], 56, 3326400, 186278400 [10, 9, 8, 7, 5, 4, 4, 3, 3, 2, 0], 48, 9979200, 479001600 [10, 9, 8, 7, 5, 4, 4, 3, 3, 1, 1], 96, 4989600, 479001600 [10, 9, 8, 7, 5, 4, 4, 3, 2, 2, 1], 164, 9979200, 1636588800 [10, 9, 8, 7, 5, 4, 4, 2, 2, 2, 2], 272, 831600, 226195200 [10, 9, 8, 7, 5, 4, 3, 3, 3, 3, 0], 80, 1663200, 133056000 [10, 9, 8, 7, 5, 4, 3, 3, 3, 2, 1], 266, 6652800, 1769644800 [10, 9, 8, 7, 5, 4, 3, 3, 2, 2, 2], 436, 3326400, 1450310400 [10, 9, 8, 7, 5, 3, 3, 3, 3, 3, 1], 424, 332640, 141039360 [10, 9, 8, 7, 5, 3, 3, 3, 3, 2, 2], 688, 831600, 572140800 [10, 9, 8, 7, 4, 4, 4, 4, 4, 1, 0], 24, 332640, 7983360 [10, 9, 8, 7, 4, 4, 4, 4, 3, 2, 0], 80, 1663200, 133056000 [10, 9, 8, 7, 4, 4, 4, 4, 3, 1, 1], 160, 831600, 133056000 [10, 9, 8, 7, 4, 4, 4, 4, 2, 2, 1], 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5, 5, 5, 4, 3, 2], 1040143136, 415800, 432491515948800 [7, 7, 6, 6, 5, 5, 5, 5, 3, 3, 3], 1498788864, 69300, 103866068275200 [7, 7, 6, 6, 5, 5, 5, 4, 4, 4, 2], 1453026384, 277200, 402778913644800 [7, 7, 6, 6, 5, 5, 5, 4, 4, 3, 3], 2086772160, 415800, 867679864128000 [7, 7, 6, 6, 5, 5, 4, 4, 4, 4, 3], 2897156288, 207900, 602318792275200 [7, 7, 6, 6, 5, 4, 4, 4, 4, 4, 4], 4011579456, 13860, 55600491260160 [7, 7, 6, 5, 5, 5, 5, 5, 5, 5, 0], 98213920, 3960, 388927123200 [7, 7, 6, 5, 5, 5, 5, 5, 5, 4, 1], 637712096, 27720, 17677379301120 [7, 7, 6, 5, 5, 5, 5, 5, 5, 3, 2], 1437662656, 27720, 39852008824320 [7, 7, 6, 5, 5, 5, 5, 5, 4, 4, 2], 2003491552, 83160, 166610357464320 [7, 7, 6, 5, 5, 5, 5, 5, 4, 3, 3], 2871790912, 83160, 238818132241920 [7, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3], 3978389984, 138600, 551404851782400 [7, 7, 6, 5, 5, 5, 4, 4, 4, 4, 4], 5497444800, 27720, 152389169856000 [7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 1], 882676480, 495, 436924857600 [7, 7, 5, 5, 5, 5, 5, 5, 5, 4, 2], 2755952416, 3960, 10913571567360 [7, 7, 5, 5, 5, 5, 5, 5, 5, 3, 3], 3942979712, 1980, 7807099829760 [7, 7, 5, 5, 5, 5, 5, 5, 4, 4, 3], 5451108032, 13860, 75552357323520 [7, 7, 5, 5, 5, 5, 5, 4, 4, 4, 4], 7517813120, 6930, 52098444921600 [7, 6, 6, 6, 6, 6, 6, 6, 5, 1, 0], 2265200, 7920, 17940384000 [7, 6, 6, 6, 6, 6, 6, 6, 4, 2, 0], 10596320, 7920, 83922854400 [7, 6, 6, 6, 6, 6, 6, 6, 4, 1, 1], 21192640, 3960, 83922854400 [7, 6, 6, 6, 6, 6, 6, 6, 3, 3, 0], 16662240, 3960, 65982470400 [7, 6, 6, 6, 6, 6, 6, 6, 3, 2, 1], 53880800, 7920, 426735936000 [7, 6, 6, 6, 6, 6, 6, 6, 2, 2, 2], 86568960, 1320, 114271027200 [7, 6, 6, 6, 6, 6, 6, 5, 5, 2, 0], 15114880, 27720, 418984473600 [7, 6, 6, 6, 6, 6, 6, 5, 5, 1, 1], 30229760, 13860, 418984473600 [7, 6, 6, 6, 6, 6, 6, 5, 4, 3, 0], 34456544, 55440, 1910270799360 [7, 6, 6, 6, 6, 6, 6, 5, 4, 2, 1], 110011616, 55440, 6099043991040 [7, 6, 6, 6, 6, 6, 6, 5, 3, 3, 1], 164094112, 27720, 4548688784640 [7, 6, 6, 6, 6, 6, 6, 5, 3, 2, 2], 259275136, 27720, 7187106769920 [7, 6, 6, 6, 6, 6, 6, 4, 4, 4, 0], 49889088, 9240, 460975173120 [7, 6, 6, 6, 6, 6, 6, 4, 4, 3, 1], 233846464, 27720, 6482223982080 [7, 6, 6, 6, 6, 6, 6, 4, 4, 2, 2], 367914752, 13860, 5099298462720 [7, 6, 6, 6, 6, 6, 6, 4, 3, 3, 2], 536998912, 27720, 14885609840640 [7, 6, 6, 6, 6, 6, 6, 3, 3, 3, 3], 779134080, 2310, 1799799724800 [7, 6, 6, 6, 6, 6, 5, 5, 5, 3, 0], 48589520, 55440, 2693802988800 [7, 6, 6, 6, 6, 6, 5, 5, 5, 2, 1], 154604560, 55440, 8571276806400 [7, 6, 6, 6, 6, 6, 5, 5, 4, 4, 0], 70116000, 83160, 5830846560000 [7, 6, 6, 6, 6, 6, 5, 5, 4, 3, 1], 326511168, 166320, 54305337461760 [7, 6, 6, 6, 6, 6, 5, 5, 4, 2, 2], 512790336, 83160, 42643644341760 [7, 6, 6, 6, 6, 6, 5, 5, 3, 3, 2], 746601312, 83160, 62087365105920 [7, 6, 6, 6, 6, 6, 5, 4, 4, 4, 1], 461792208, 55440, 25601760011520 [7, 6, 6, 6, 6, 6, 5, 4, 4, 3, 2], 1047380832, 166320, 174200379978240 [7, 6, 6, 6, 6, 6, 5, 4, 3, 3, 3], 1510652416, 55440, 83750569943040 [7, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2], 1464314240, 13860, 20295395366400 [7, 6, 6, 6, 6, 6, 4, 4, 4, 3, 3], 2104822240, 27720, 58345672492800 [7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 0], 98254848, 69300, 6809060966400 [7, 6, 6, 6, 6, 5, 5, 5, 5, 3, 1], 454539776, 69300, 31499606476800 [7, 6, 6, 6, 6, 5, 5, 5, 5, 2, 2], 712569856, 34650, 24690545510400 [7, 6, 6, 6, 6, 5, 5, 5, 4, 4, 1], 641030720, 138600, 88846857792000 [7, 6, 6, 6, 6, 5, 5, 5, 4, 3, 2], 1448074944, 277200, 401406374476800 [7, 6, 6, 6, 6, 5, 5, 5, 3, 3, 3], 2084249376, 46200, 96292321171200 [7, 6, 6, 6, 6, 5, 5, 4, 4, 4, 2], 2019492864, 138600, 279901710950400 [7, 6, 6, 6, 6, 5, 5, 4, 4, 3, 3], 2897156288, 207900, 602318792275200 [7, 6, 6, 6, 6, 5, 4, 4, 4, 4, 3], 4015912704, 69300, 278302750387200 [7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4], 5552231680, 2310, 12825655180800 [7, 6, 6, 6, 5, 5, 5, 5, 5, 5, 0], 137311360, 9240, 1268756966400 [7, 6, 6, 6, 5, 5, 5, 5, 5, 4, 1], 887523120, 55440, 49204281772800 [7, 6, 6, 6, 5, 5, 5, 5, 5, 3, 2], 1996969200, 55440, 110711972448000 [7, 6, 6, 6, 5, 5, 5, 5, 4, 4, 2], 2778463264, 138600, 385095008390400 [7, 6, 6, 6, 5, 5, 5, 5, 4, 3, 3], 3978389984, 138600, 551404851782400 [7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3], 5503086136, 184800, 1016970317932800 [7, 6, 6, 6, 5, 5, 4, 4, 4, 4, 4], 7593232160, 27720, 210484395475200 [7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 1], 1225800800, 3960, 4854171168000 [7, 6, 6, 5, 5, 5, 5, 5, 5, 4, 2], 3814043456, 27720, 105725284600320 [7, 6, 6, 5, 5, 5, 5, 5, 5, 3, 3], 5451108032, 13860, 75552357323520 [7, 6, 6, 5, 5, 5, 5, 5, 4, 4, 3], 7525157184, 83160, 625792071421440 [7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4], 10363669504, 34650, 359101148313600 [7, 6, 5, 5, 5, 5, 5, 5, 5, 5, 2], 5224459520, 990, 5172214924800 [7, 6, 5, 5, 5, 5, 5, 5, 5, 4, 3], 10269804048, 7920, 81336848060160 [7, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4], 14118032288, 9240, 130450618341120 [7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3], 13989118080, 110, 1538802988800 [7, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4], 19197704960, 495, 9502863955200 [6, 6, 6, 6, 6, 6, 6, 6, 6, 1, 0], 3230080, 110, 355308800 [6, 6, 6, 6, 6, 6, 6, 6, 5, 2, 0], 21351680, 990, 21138163200 [6, 6, 6, 6, 6, 6, 6, 6, 5, 1, 1], 42703360, 495, 21138163200 [6, 6, 6, 6, 6, 6, 6, 6, 4, 3, 0], 48482560, 990, 47997734400 [6, 6, 6, 6, 6, 6, 6, 6, 4, 2, 1], 154604800, 990, 153058752000 [6, 6, 6, 6, 6, 6, 6, 6, 3, 3, 1], 230263040, 495, 113980204800 [6, 6, 6, 6, 6, 6, 6, 6, 3, 2, 2], 363560960, 495, 179962675200 [6, 6, 6, 6, 6, 6, 6, 5, 5, 3, 0], 68193440, 3960, 270046022400 [6, 6, 6, 6, 6, 6, 6, 5, 5, 2, 1], 216700960, 3960, 858135801600 [6, 6, 6, 6, 6, 6, 6, 5, 4, 4, 0], 98213920, 3960, 388927123200 [6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 1], 456085728, 7920, 3612198965760 [6, 6, 6, 6, 6, 6, 6, 5, 4, 2, 2], 715743616, 3960, 2834344719360 [6, 6, 6, 6, 6, 6, 6, 5, 3, 3, 2], 1040788672, 3960, 4121523141120 [6, 6, 6, 6, 6, 6, 6, 4, 4, 4, 1], 643865376, 1320, 849902296320 [6, 6, 6, 6, 6, 6, 6, 4, 4, 3, 2], 1457477504, 3960, 5771610915840 [6, 6, 6, 6, 6, 6, 6, 4, 3, 3, 3], 2099748672, 1320, 2771668247040 [6, 6, 6, 6, 6, 6, 5, 5, 5, 4, 0], 137311360, 9240, 1268756966400 [6, 6, 6, 6, 6, 6, 5, 5, 5, 3, 1], 633428800, 9240, 5852882112000 [6, 6, 6, 6, 6, 6, 5, 5, 5, 2, 2], 992234880, 4620, 4584125145600 [6, 6, 6, 6, 6, 6, 5, 5, 4, 4, 1], 891746560, 13860, 12359607321600 [6, 6, 6, 6, 6, 6, 5, 5, 4, 3, 2], 2010510848, 27720, 55731360706560 [6, 6, 6, 6, 6, 6, 5, 5, 3, 3, 3], 2890550976, 4620, 13354345509120 [6, 6, 6, 6, 6, 6, 5, 4, 4, 4, 2], 2799322368, 9240, 25865738680320 [6, 6, 6, 6, 6, 6, 5, 4, 4, 3, 3], 4011579456, 13860, 55600491260160 [6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 3], 5552231680, 2310, 12825655180800 [6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 0], 191474240, 2772, 530766593280 [6, 6, 6, 6, 6, 5, 5, 5, 5, 4, 1], 1231993920, 13860, 17075435731200 [6, 6, 6, 6, 6, 5, 5, 5, 5, 3, 2], 2766671680, 13860, 38346069484800 [6, 6, 6, 6, 6, 5, 5, 5, 4, 4, 2], 3843404960, 27720, 106539185491200 [6, 6, 6, 6, 6, 5, 5, 5, 4, 3, 3], 5497444800, 27720, 152389169856000 [6, 6, 6, 6, 6, 5, 5, 4, 4, 4, 3], 7593232160, 27720, 210484395475200 [6, 6, 6, 6, 6, 5, 4, 4, 4, 4, 4], 10462509760, 2772, 29002077054720 [6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 1], 1698090880, 2310, 3922589932800 [6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 2], 5265523200, 13860, 72980151552000 [6, 6, 6, 6, 5, 5, 5, 5, 5, 3, 3], 7517813120, 6930, 52098444921600 [6, 6, 6, 6, 5, 5, 5, 5, 4, 4, 3], 10363669504, 34650, 359101148313600 [6, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4], 14253550336, 11550, 164628506380800 [6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 2], 7199139360, 1320, 9502863955200 [6, 6, 6, 5, 5, 5, 5, 5, 5, 4, 3], 14118032288, 9240, 130450618341120 [6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4], 19382885376, 9240, 179097860874240 [6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 3], 19197704960, 495, 9502863955200 [6, 6, 5, 5, 5, 5, 5, 5, 5, 4, 4], 26312345920, 1980, 52098444921600 [6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4], 35659908480, 110, 3922589932800 [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], 48251508480, 1, 48251508480 Sum of numbers in last column=, 36028797018963968 The most probable vector is, [7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3], which arises in, 1016970317932800, out of 2 to the power, 55, that equals, 36028797018963968, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 11, players The sole champion score-vector is, [7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3] the corresponding record is, 1016970317932800 ----------------------------------------------------------------------------\ ----- 12 / 12 \ --------' | --------' | ' | | |' | | | 12, Players. Symmetric Function:, | | | | | (a[i] + a[j])| | | | | | | | | | | | | i = 1 \j = i + 1 / Score Vector, Coeff. of Sym. Func. , Number of Terms in Sym. Funct., Events in Tournaments. [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], 1, 479001600, 479001600 [11, 10, 9, 8, 7, 6, 5, 4, 3, 1, 1, 1], 2, 79833600, 159667200 [11, 10, 9, 8, 7, 6, 5, 4, 2, 2, 2, 0], 2, 79833600, 159667200 [11, 10, 9, 8, 7, 6, 5, 4, 2, 2, 1, 1], 4, 119750400, 479001600 [11, 10, 9, 8, 7, 6, 5, 3, 3, 3, 1, 0], 2, 79833600, 159667200 [11, 10, 9, 8, 7, 6, 5, 3, 3, 2, 2, 0], 4, 119750400, 479001600 [11, 10, 9, 8, 7, 6, 5, 3, 3, 2, 1, 1], 8, 119750400, 958003200 [11, 10, 9, 8, 7, 6, 5, 3, 2, 2, 2, 1], 14, 79833600, 1117670400 [11, 10, 9, 8, 7, 6, 5, 2, 2, 2, 2, 2], 24, 3991680, 95800320 [11, 10, 9, 8, 7, 6, 4, 4, 4, 2, 1, 0], 2, 79833600, 159667200 [11, 10, 9, 8, 7, 6, 4, 4, 4, 1, 1, 1], 4, 13305600, 53222400 [11, 10, 9, 8, 7, 6, 4, 4, 3, 3, 1, 0], 4, 119750400, 479001600 [11, 10, 9, 8, 7, 6, 4, 4, 3, 2, 2, 0], 8, 119750400, 958003200 [11, 10, 9, 8, 7, 6, 4, 4, 3, 2, 1, 1], 16, 119750400, 1916006400 [11, 10, 9, 8, 7, 6, 4, 4, 2, 2, 2, 1], 28, 39916800, 1117670400 [11, 10, 9, 8, 7, 6, 4, 3, 3, 3, 2, 0], 14, 79833600, 1117670400 [11, 10, 9, 8, 7, 6, 4, 3, 3, 3, 1, 1], 28, 39916800, 1117670400 [11, 10, 9, 8, 7, 6, 4, 3, 3, 2, 2, 1], 48, 119750400, 5748019200 [11, 10, 9, 8, 7, 6, 4, 3, 2, 2, 2, 2], 80, 19958400, 1596672000 [11, 10, 9, 8, 7, 6, 3, 3, 3, 3, 3, 0], 24, 3991680, 95800320 [11, 10, 9, 8, 7, 6, 3, 3, 3, 3, 2, 1], 80, 19958400, 1596672000 [11, 10, 9, 8, 7, 6, 3, 3, 3, 2, 2, 2], 132, 13305600, 1756339200 [11, 10, 9, 8, 7, 5, 5, 5, 3, 2, 1, 0], 2, 79833600, 159667200 [11, 10, 9, 8, 7, 5, 5, 5, 3, 1, 1, 1], 4, 13305600, 53222400 [11, 10, 9, 8, 7, 5, 5, 5, 2, 2, 2, 0], 4, 13305600, 53222400 [11, 10, 9, 8, 7, 5, 5, 5, 2, 2, 1, 1], 8, 19958400, 159667200 [11, 10, 9, 8, 7, 5, 5, 4, 4, 2, 1, 0], 4, 119750400, 479001600 [11, 10, 9, 8, 7, 5, 5, 4, 4, 1, 1, 1], 8, 19958400, 159667200 [11, 10, 9, 8, 7, 5, 5, 4, 3, 3, 1, 0], 8, 119750400, 958003200 [11, 10, 9, 8, 7, 5, 5, 4, 3, 2, 2, 0], 16, 119750400, 1916006400 [11, 10, 9, 8, 7, 5, 5, 4, 3, 2, 1, 1], 32, 119750400, 3832012800 [11, 10, 9, 8, 7, 5, 5, 4, 2, 2, 2, 1], 56, 39916800, 2235340800 [11, 10, 9, 8, 7, 5, 5, 3, 3, 3, 2, 0], 28, 39916800, 1117670400 [11, 10, 9, 8, 7, 5, 5, 3, 3, 3, 1, 1], 56, 19958400, 1117670400 [11, 10, 9, 8, 7, 5, 5, 3, 3, 2, 2, 1], 96, 59875200, 5748019200 [11, 10, 9, 8, 7, 5, 5, 3, 2, 2, 2, 2], 160, 9979200, 1596672000 [11, 10, 9, 8, 7, 5, 4, 4, 4, 3, 1, 0], 14, 79833600, 1117670400 [11, 10, 9, 8, 7, 5, 4, 4, 4, 2, 2, 0], 28, 39916800, 1117670400 [11, 10, 9, 8, 7, 5, 4, 4, 4, 2, 1, 1], 56, 39916800, 2235340800 [11, 10, 9, 8, 7, 5, 4, 4, 3, 3, 2, 0], 48, 119750400, 5748019200 [11, 10, 9, 8, 7, 5, 4, 4, 3, 3, 1, 1], 96, 59875200, 5748019200 [11, 10, 9, 8, 7, 5, 4, 4, 3, 2, 2, 1], 164, 119750400, 19639065600 [11, 10, 9, 8, 7, 5, 4, 4, 2, 2, 2, 2], 272, 9979200, 2714342400 [11, 10, 9, 8, 7, 5, 4, 3, 3, 3, 3, 0], 80, 19958400, 1596672000 [11, 10, 9, 8, 7, 5, 4, 3, 3, 3, 2, 1], 266, 79833600, 21235737600 [11, 10, 9, 8, 7, 5, 4, 3, 3, 2, 2, 2], 436, 39916800, 17403724800 [11, 10, 9, 8, 7, 5, 3, 3, 3, 3, 3, 1], 424, 3991680, 1692472320 [11, 10, 9, 8, 7, 5, 3, 3, 3, 3, 2, 2], 688, 9979200, 6865689600 [11, 10, 9, 8, 7, 4, 4, 4, 4, 4, 1, 0], 24, 3991680, 95800320 [11, 10, 9, 8, 7, 4, 4, 4, 4, 3, 2, 0], 80, 19958400, 1596672000 [11, 10, 9, 8, 7, 4, 4, 4, 4, 3, 1, 1], 160, 9979200, 1596672000 [11, 10, 9, 8, 7, 4, 4, 4, 4, 2, 2, 1], 272, 9979200, 2714342400 [11, 10, 9, 8, 7, 4, 4, 4, 3, 3, 3, 0], 132, 13305600, 1756339200 [11, 10, 9, 8, 7, 4, 4, 4, 3, 3, 2, 1], 436, 39916800, 17403724800 [11, 10, 9, 8, 7, 4, 4, 4, 3, 2, 2, 2], 712, 13305600, 9473587200 [11, 10, 9, 8, 7, 4, 4, 3, 3, 3, 3, 1], 688, 9979200, 6865689600 [11, 10, 9, 8, 7, 4, 4, 3, 3, 3, 2, 2], 1112, 19958400, 22193740800 [11, 10, 9, 8, 7, 4, 3, 3, 3, 3, 3, 2], 1720, 3991680, 6865689600 [11, 10, 9, 8, 7, 3, 3, 3, 3, 3, 3, 3], 2640, 95040, 250905600 [11, 10, 9, 8, 6, 6, 6, 4, 3, 2, 1, 0], 2, 79833600, 159667200 [11, 10, 9, 8, 6, 6, 6, 4, 3, 1, 1, 1], 4, 13305600, 53222400 [11, 10, 9, 8, 6, 6, 6, 4, 2, 2, 2, 0], 4, 13305600, 53222400 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1], 3396181760, 13860, 47071079193600 [7, 7, 7, 7, 6, 6, 6, 6, 6, 5, 3, 0], 5265523200, 166320, 875761818624000 [7, 7, 7, 7, 6, 6, 6, 6, 6, 5, 2, 1], 16673676160, 166320, 2773165818931200 [7, 7, 7, 7, 6, 6, 6, 6, 6, 4, 4, 0], 7517813120, 83160, 625181339059200 [7, 7, 7, 7, 6, 6, 6, 6, 6, 4, 3, 1], 34604571520, 166320, 5755432335206400 [7, 7, 7, 7, 6, 6, 6, 6, 6, 4, 2, 2], 54173516800, 83160, 4505069657088000 [7, 7, 7, 7, 6, 6, 6, 6, 6, 3, 3, 2], 78395863040, 83160, 6519399970406400 [7, 7, 7, 7, 6, 6, 6, 6, 5, 5, 4, 0], 10363669504, 415800, 4309213779763200 [7, 7, 7, 7, 6, 6, 6, 6, 5, 5, 3, 1], 47382188800, 415800, 19701514103040000 [7, 7, 7, 7, 6, 6, 6, 6, 5, 5, 2, 2], 74037038592, 207900, 15392300323276800 [7, 7, 7, 7, 6, 6, 6, 6, 5, 4, 4, 1], 66148491264, 415800, 27504542667571200 [7, 7, 7, 7, 6, 6, 6, 6, 5, 4, 3, 2], 148065504384, 831600, 123131273445734400 [7, 7, 7, 7, 6, 6, 6, 6, 5, 3, 3, 3], 211897677568, 138600, 29369018110924800 [7, 7, 7, 7, 6, 6, 6, 6, 4, 4, 4, 2], 204485054976, 138600, 28341628619673600 [7, 7, 7, 7, 6, 6, 6, 6, 4, 4, 3, 3], 291708753664, 207900, 60646249886745600 [7, 7, 7, 7, 6, 6, 6, 5, 5, 5, 5, 0], 14253550336, 138600, 1975542076569600 [7, 7, 7, 7, 6, 6, 6, 5, 5, 5, 4, 1], 90136599872, 554400, 49971730969036800 [7, 7, 7, 7, 6, 6, 6, 5, 5, 5, 3, 2], 200961862720, 554400, 111413256691968000 [7, 7, 7, 7, 6, 6, 6, 5, 5, 4, 4, 2], 276943158976, 831600, 230305931004441600 [7, 7, 7, 7, 6, 6, 6, 5, 5, 4, 3, 3], 394340457216, 831600, 327933524220825600 [7, 7, 7, 7, 6, 6, 6, 5, 4, 4, 4, 3], 540412613184, 554400, 299604752749209600 [7, 7, 7, 7, 6, 6, 6, 4, 4, 4, 4, 4], 738857086080, 27720, 20481118426137600 [7, 7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 1], 122565699840, 83160, 10192563598694400 [7, 7, 7, 7, 6, 6, 5, 5, 5, 5, 4, 2], 374351873024, 415800, 155655508803379200 [7, 7, 7, 7, 6, 6, 5, 5, 5, 5, 3, 3], 532079447040, 207900, 110619317039616000 [7, 7, 7, 7, 6, 6, 5, 5, 5, 4, 4, 3], 727835295104, 831600, 605267831408486400 [7, 7, 7, 7, 6, 6, 5, 5, 4, 4, 4, 4], 993370548224, 207900, 206521736975769600 [7, 7, 7, 7, 6, 5, 5, 5, 5, 5, 5, 2], 505094288896, 27720, 14001213688197120 [7, 7, 7, 7, 6, 5, 5, 5, 5, 5, 4, 3], 978588046720, 166320, 162758763930470400 [7, 7, 7, 7, 6, 5, 5, 5, 5, 4, 4, 4], 1333374989184, 138600, 184805773500902400 [7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 3], 1313591095040, 3960, 5201820736358400 [7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 4, 4], 1786961582848, 13860, 24767287538273280 [7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 3, 0], 7199139360, 15840, 114034367462400 [7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 2, 1], 22763177920, 15840, 360568738252800 [7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 4, 0], 14118032288, 110880, 1565407420093440 [7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 3, 1], 64352441216, 110880, 7135398682030080 [7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 2, 2], 100468817856, 55440, 5569991261936640 [7, 7, 7, 6, 6, 6, 6, 6, 6, 4, 4, 1], 89696267392, 55440, 4972761064212480 [7, 7, 7, 6, 6, 6, 6, 6, 6, 4, 3, 2], 200372013440, 110880, 22217248850227200 [7, 7, 7, 6, 6, 6, 6, 6, 6, 3, 3, 3], 286437423168, 18480, 5293363580144640 [7, 7, 7, 6, 6, 6, 6, 6, 5, 5, 5, 0], 19382885376, 110880, 2149174330490880 [7, 7, 7, 6, 6, 6, 6, 6, 5, 5, 4, 1], 122012769024, 332640, 40586327488143360 [7, 7, 7, 6, 6, 6, 6, 6, 5, 5, 3, 2], 271489988960, 332640, 90308429927654400 [7, 7, 7, 6, 6, 6, 6, 6, 5, 4, 4, 2], 373614648288, 332640, 124279176606520320 [7, 7, 7, 6, 6, 6, 6, 6, 5, 4, 3, 3], 531439823104, 332640, 176778142757314560 [7, 7, 7, 6, 6, 6, 6, 6, 4, 4, 4, 3], 727350823968, 110880, 80648659361571840 [7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 1], 165636751360, 138600, 22957253738496000 [7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 4, 2], 504230008096, 554400, 279545116488422400 [7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 3, 3], 715952284224, 277200, 198461973186892800 [7, 7, 7, 6, 6, 6, 6, 5, 5, 4, 4, 3], 978132041600, 831600, 813414605794560000 [7, 7, 7, 6, 6, 6, 6, 5, 4, 4, 4, 4], 1333374989184, 138600, 184805773500902400 [7, 7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 2], 679306104480, 110880, 75321460864742400 [7, 7, 7, 6, 6, 6, 5, 5, 5, 5, 4, 3], 1313218581600, 554400, 728048381639040000 [7, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4], 1787244174480, 369600, 660565446887808000 [7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 5, 3], 1760331490752, 55440, 97592777847290880 [7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 4, 4], 2391995707904, 166320, 397836726138593280 [7, 7, 7, 6, 5, 5, 5, 5, 5, 5, 5, 4], 3196749970400, 15840, 50636519531136000 [7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5], 4266318330624, 220, 938590032737280 [7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 4, 0], 19197704960, 5940, 114034367462400 [7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 3, 1], 87239738880, 5940, 518204048947200 [7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2], 136084067840, 2970, 404169681484800 [7, 7, 6, 6, 6, 6, 6, 6, 6, 5, 5, 0], 26312345920, 23760, 625181339059200 [7, 7, 6, 6, 6, 6, 6, 6, 6, 5, 4, 1], 164875884992, 47520, 7834902054819840 [7, 7, 6, 6, 6, 6, 6, 6, 6, 5, 3, 2], 366134981632, 47520, 17398734327152640 [7, 7, 6, 6, 6, 6, 6, 6, 6, 4, 4, 2], 503176581824, 23760, 11955475584138240 [7, 7, 6, 6, 6, 6, 6, 6, 6, 4, 3, 3], 714996803584, 23760, 16988324053155840 [7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 1], 223470414592, 55440, 12389199784980480 [7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 4, 2], 678059414080, 166320, 112774841749785600 [7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 3, 3], 961802690816, 83160, 79983511768258560 [7, 7, 6, 6, 6, 6, 6, 6, 5, 4, 4, 3], 1312412213888, 166320, 218280399413852160 [7, 7, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4], 1786961582848, 13860, 24767287538273280 [7, 7, 6, 6, 6, 6, 6, 5, 5, 5, 5, 2], 912166052864, 83160, 75855728956170240 [7, 7, 6, 6, 6, 6, 6, 5, 5, 5, 4, 3], 1759569656192, 332640, 585303250435706880 [7, 7, 6, 6, 6, 6, 6, 5, 5, 4, 4, 4], 2391995707904, 166320, 397836726138593280 [7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 5, 3], 2355500480000, 83160, 195883419916800000 [7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 4], 3197215017984, 207900, 664701002238873600 [7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 5, 4], 4267511444608, 55440, 236590834489067520 [7, 7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5], 5688424440832, 2970, 16894620589271040 [7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 0], 35659908480, 1320, 47071079193600 [7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 1], 222428371200, 1320, 293605449984000 [7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 3, 2], 492957615360, 1320, 650704052275200 [7, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 1], 301016289280, 5940, 1788036758323200 [7, 6, 6, 6, 6, 6, 6, 6, 6, 5, 4, 2], 910390475008, 11880, 10815438843095040 [7, 6, 6, 6, 6, 6, 6, 6, 6, 5, 3, 3], 1290068188416, 5940, 7663005039191040 [7, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 3], 1758251704832, 5940, 10444015126702080 [7, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 2], 1222998923552, 15840, 19372302949063680 [7, 6, 6, 6, 6, 6, 6, 6, 5, 5, 4, 3], 2354159379264, 47520, 111869653702625280 [7, 6, 6, 6, 6, 6, 6, 6, 5, 4, 4, 4], 3196749970400, 15840, 50636519531136000 [7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 3], 3147402263552, 27720, 87245990745661440 [7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 4, 4], 4267511444608, 55440, 236590834489067520 [7, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4], 5689189902400, 33264, 189245212913433600 [7, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5], 7574562746240, 3960, 29995268475110400 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0], 48251508480, 12, 579018101760 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 1], 404850593280, 132, 53440278312960 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 2], 1220495808000, 132, 161105446656000 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 3, 3], 1727793446400, 66, 114034367462400 [6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 2], 1637374314240, 660, 1080667047398400 [6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3], 3145215720192, 1320, 4151684750653440 [6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 4], 4266318330624, 220, 938590032737280 [6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 3], 4199767950336, 1980, 8315540541665280 [6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 4, 4], 5688424440832, 2970, 16894620589271040 [6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 4], 7574562746240, 3960, 29995268475110400 [6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5], 10073269059840, 924, 9307700611292160 Sum of numbers in last column=, 73786976294838206464 The most probable vector is, [8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 3], which arises in, 1772428331094220800, out of 2 to the power, 66, that equals, 73786976294838206464, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 12, players The sole champion score-vector is, [8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 3] the corresponding record is, 1772428331094220800 ------------------------------------------ In conclusion, the n=2 thorough n=, 12, of OEIS sequence A274098 are [6, 24, 280, 8640, 233520, 23157120, 5329376640, 1314169920000, 1016970317932800, 1772428331094220800] ------------------------------------------ This ends this article, that took, 11.388, seconds to produce.