All the Possible Score Vectors in a Round-Robin Soccer Tournament, and the N\ umber of Times each Shows Up for tournaments from Three to , 7, players. By Shalosh B. Ekhad In his beautiful articles, "An American Tournament Treated by the Calculus o\ f Symmetric Functions" Quarterly J. of Pure and Applied Mathematics. vol. XLIX, No. 193, 1920 and "Chess Tournament and the like treated by the calculus of Symmetric Functio\ ns" 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 Tennis and Chess tournaments, respectively, for n pla\ yers for n up to 9 and 6 respectively, and for each found all the possible score vectors and for each their number of occurrences. In this article, we do the analogous thing for Soccer for tournments up to, 7, players. Recall that in a Soccer match with a winner, the winner gets three point, an\ d the loser gets 0 points, and in case of a draw, each gets one point. ----------------------------------------------------------------------------\ ----- 3, Players. Symmetric Function:, 3 / 3 \ --------' | --------' | ' | | |' | | 3 3 | | | | | | (a[i] + a[j] + 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, 2, 2], 1, 1, 1 [4, 2, 1], 1, 6, 6 [6, 1, 1], 1, 3, 3 [4, 4, 0], 1, 3, 3 [4, 3, 1], 1, 6, 6 [6, 3, 0], 1, 6, 6 [3, 3, 3], 2, 1, 2 Sum of numbers in last column=, 27 The most probable vector is, [4, 2, 1], which arises in, 6, out of, 3, to the power, 3 that equals, 27, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 3, players The set of champion score vectors is, {[4, 2, 1], [4, 3, 1], [6, 3, 0]} the corresponding record is, 6 ----------------------------------------------------------------------------\ ----- 4, Players. Symmetric Function:, 4 / 4 \ --------' | --------' | ' | | |' | | 3 3 | | | | | | (a[i] + a[j] + 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, 3, 3, 3], 1, 1, 1 [5, 3, 3, 2], 1, 12, 12 [7, 3, 2, 2], 1, 12, 12 [5, 5, 3, 1], 1, 12, 12 [5, 5, 2, 2], 2, 6, 12 [5, 4, 3, 2], 1, 24, 24 [9, 2, 2, 2], 1, 4, 4 [7, 5, 2, 1], 1, 24, 24 [7, 4, 3, 1], 1, 24, 24 [7, 4, 2, 2], 2, 12, 24 [6, 5, 2, 2], 1, 12, 12 [5, 5, 5, 0], 1, 4, 4 [5, 5, 4, 1], 2, 12, 24 [5, 5, 3, 2], 1, 12, 12 [5, 4, 4, 2], 2, 12, 24 [4, 4, 4, 3], 2, 4, 8 [9, 4, 2, 1], 1, 24, 24 [7, 7, 1, 1], 1, 6, 6 [7, 6, 2, 1], 1, 24, 24 [7, 5, 4, 0], 1, 24, 24 [7, 5, 3, 1], 1, 24, 24 [7, 4, 4, 1], 3, 12, 36 [7, 4, 3, 2], 1, 24, 24 [6, 5, 4, 1], 1, 24, 24 [6, 4, 4, 2], 2, 12, 24 [5, 4, 4, 3], 2, 12, 24 [4, 4, 4, 4], 6, 1, 6 [9, 6, 1, 1], 1, 12, 12 [9, 4, 4, 0], 1, 12, 12 [9, 4, 3, 1], 1, 24, 24 [7, 7, 3, 0], 1, 12, 12 [7, 6, 4, 0], 1, 24, 24 [7, 6, 3, 1], 1, 24, 24 [7, 4, 3, 3], 2, 12, 24 [6, 6, 4, 1], 2, 12, 24 [6, 4, 4, 3], 3, 12, 36 [9, 6, 3, 0], 1, 24, 24 [9, 3, 3, 3], 2, 4, 8 [6, 6, 6, 0], 2, 4, 8 [6, 6, 3, 3], 4, 6, 24 Sum of numbers in last column=, 729 The most probable vector is, [6, 4, 4, 3], which arises in, 36, out of, 3, to the power, 6 that equals, 729, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 4, players The set of champion score vectors is, {[6, 4, 4, 3], [7, 4, 4, 1]} the corresponding record is, 36 ----------------------------------------------------------------------------\ ----- 5, Players. Symmetric Function:, 5 / 5 \ --------' | --------' | ' | | |' | | 3 3 | | | | | | (a[i] + a[j] + 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, 4, 4, 4, 4], 1, 1, 1 [6, 4, 4, 4, 3], 1, 20, 20 [8, 4, 4, 3, 3], 1, 30, 30 [6, 6, 4, 4, 2], 1, 30, 30 [6, 6, 4, 3, 3], 2, 30, 60 [6, 5, 4, 4, 3], 1, 60, 60 [10, 4, 3, 3, 3], 1, 20, 20 [8, 6, 4, 3, 2], 1, 120, 120 [8, 6, 3, 3, 3], 3, 20, 60 [8, 5, 4, 4, 2], 1, 60, 60 [8, 5, 4, 3, 3], 2, 60, 120 [7, 6, 4, 3, 3], 1, 60, 60 [6, 6, 6, 4, 1], 1, 20, 20 [6, 6, 6, 3, 2], 3, 20, 60 [6, 6, 5, 4, 2], 2, 60, 120 [6, 6, 5, 3, 3], 4, 30, 120 [6, 6, 4, 4, 3], 2, 30, 60 [6, 5, 5, 4, 3], 2, 60, 120 [5, 5, 5, 4, 4], 2, 10, 20 [12, 3, 3, 3, 3], 1, 5, 5 [10, 6, 3, 3, 2], 1, 60, 60 [10, 5, 4, 3, 2], 1, 120, 120 [10, 5, 3, 3, 3], 3, 20, 60 [9, 6, 3, 3, 3], 1, 20, 20 [8, 8, 4, 2, 2], 1, 30, 30 [8, 8, 3, 3, 2], 2, 30, 60 [8, 7, 4, 3, 2], 1, 120, 120 [8, 7, 3, 3, 3], 3, 20, 60 [8, 6, 6, 3, 1], 1, 60, 60 [8, 6, 6, 2, 2], 2, 30, 60 [8, 6, 5, 4, 1], 1, 120, 120 [8, 6, 5, 3, 2], 4, 120, 480 [8, 6, 4, 4, 2], 2, 60, 120 [8, 6, 4, 3, 3], 2, 60, 120 [8, 5, 5, 4, 2], 3, 60, 180 [8, 5, 5, 3, 3], 6, 30, 180 [8, 5, 4, 4, 3], 2, 60, 120 [7, 6, 6, 3, 2], 2, 60, 120 [7, 6, 5, 4, 2], 1, 120, 120 [7, 6, 5, 3, 3], 3, 60, 180 [7, 5, 5, 4, 3], 2, 60, 120 [6, 6, 6, 6, 0], 1, 5, 5 [6, 6, 6, 5, 1], 3, 20, 60 [6, 6, 6, 4, 2], 3, 20, 60 [6, 6, 6, 3, 3], 5, 10, 50 [6, 6, 5, 5, 2], 6, 30, 180 [6, 6, 5, 4, 3], 3, 60, 180 [6, 5, 5, 5, 3], 8, 20, 160 [6, 5, 5, 4, 4], 4, 30, 120 [5, 5, 5, 5, 4], 6, 5, 30 [12, 5, 3, 3, 2], 1, 60, 60 [10, 8, 3, 2, 2], 1, 60, 60 [10, 7, 4, 2, 2], 1, 60, 60 [10, 7, 3, 3, 2], 2, 60, 120 [10, 6, 5, 3, 1], 1, 120, 120 [10, 6, 5, 2, 2], 2, 60, 120 [10, 6, 4, 3, 2], 1, 120, 120 [10, 5, 5, 4, 1], 1, 60, 60 [10, 5, 5, 3, 2], 5, 60, 300 [10, 5, 4, 4, 2], 2, 60, 120 [10, 5, 4, 3, 3], 2, 60, 120 [9, 8, 3, 3, 2], 1, 60, 60 [9, 6, 5, 3, 2], 1, 120, 120 [9, 5, 5, 3, 3], 2, 30, 60 [8, 8, 6, 2, 1], 1, 60, 60 [8, 8, 5, 3, 1], 2, 60, 120 [8, 8, 5, 2, 2], 4, 30, 120 [8, 8, 4, 4, 1], 2, 30, 60 [8, 8, 4, 3, 2], 3, 60, 180 [8, 7, 6, 3, 1], 1, 120, 120 [8, 7, 6, 2, 2], 3, 60, 180 [8, 7, 5, 4, 1], 1, 120, 120 [8, 7, 5, 3, 2], 5, 120, 600 [8, 7, 4, 4, 2], 2, 60, 120 [8, 7, 4, 3, 3], 2, 60, 120 [8, 6, 6, 5, 0], 1, 60, 60 [8, 6, 6, 4, 1], 2, 60, 120 [8, 6, 6, 3, 2], 3, 60, 180 [8, 6, 5, 5, 1], 5, 60, 300 [8, 6, 5, 4, 2], 5, 120, 600 [8, 6, 5, 3, 3], 3, 60, 180 [8, 6, 4, 4, 3], 2, 60, 120 [8, 5, 5, 5, 2], 12, 20, 240 [8, 5, 5, 4, 3], 7, 60, 420 [8, 5, 4, 4, 4], 6, 20, 120 [7, 7, 6, 3, 2], 2, 60, 120 [7, 7, 5, 4, 2], 2, 60, 120 [7, 7, 5, 3, 3], 4, 30, 120 [7, 6, 6, 5, 1], 2, 60, 120 [7, 6, 6, 4, 2], 2, 60, 120 [7, 6, 5, 5, 2], 7, 60, 420 [7, 6, 5, 4, 3], 3, 120, 360 [7, 5, 5, 5, 3], 12, 20, 240 [7, 5, 5, 4, 4], 6, 30, 180 [6, 6, 6, 5, 2], 3, 20, 60 [6, 6, 5, 5, 3], 6, 30, 180 [6, 6, 5, 4, 4], 4, 30, 120 [6, 5, 5, 5, 4], 12, 20, 240 [5, 5, 5, 5, 5], 24, 1, 24 [12, 7, 3, 2, 2], 1, 60, 60 [12, 5, 5, 3, 1], 1, 60, 60 [12, 5, 5, 2, 2], 2, 30, 60 [12, 5, 4, 3, 2], 1, 120, 120 [10, 10, 2, 2, 2], 1, 10, 10 [10, 9, 3, 2, 2], 1, 60, 60 [10, 8, 5, 2, 1], 1, 120, 120 [10, 8, 4, 3, 1], 1, 120, 120 [10, 8, 4, 2, 2], 2, 60, 120 [10, 7, 6, 2, 1], 1, 120, 120 [10, 7, 5, 3, 1], 2, 120, 240 [10, 7, 5, 2, 2], 5, 60, 300 [10, 7, 4, 4, 1], 2, 60, 120 [10, 7, 4, 3, 2], 3, 120, 360 [10, 6, 6, 2, 2], 2, 30, 60 [10, 6, 5, 5, 0], 1, 60, 60 [10, 6, 5, 4, 1], 2, 120, 240 [10, 6, 5, 3, 2], 2, 120, 240 [10, 6, 4, 4, 2], 2, 60, 120 [10, 5, 5, 5, 1], 7, 20, 140 [10, 5, 5, 4, 2], 7, 60, 420 [10, 5, 5, 3, 3], 2, 30, 60 [10, 5, 4, 4, 3], 4, 60, 240 [10, 4, 4, 4, 4], 8, 5, 40 [9, 8, 5, 3, 1], 1, 120, 120 [9, 8, 5, 2, 2], 2, 60, 120 [9, 8, 4, 3, 2], 1, 120, 120 [9, 7, 6, 2, 2], 1, 60, 60 [9, 7, 5, 3, 2], 2, 120, 240 [9, 6, 5, 5, 1], 1, 60, 60 [9, 6, 5, 4, 2], 1, 120, 120 [9, 5, 5, 5, 2], 6, 20, 120 [9, 5, 5, 4, 3], 3, 60, 180 [8, 8, 8, 1, 1], 1, 10, 10 [8, 8, 7, 2, 1], 2, 60, 120 [8, 8, 6, 4, 0], 1, 60, 60 [8, 8, 6, 3, 1], 2, 60, 120 [8, 8, 6, 2, 2], 2, 30, 60 [8, 8, 5, 5, 0], 2, 30, 60 [8, 8, 5, 4, 1], 5, 60, 300 [8, 8, 5, 3, 2], 2, 60, 120 [8, 8, 4, 4, 2], 5, 30, 150 [8, 8, 4, 3, 3], 2, 30, 60 [8, 7, 7, 3, 1], 2, 60, 120 [8, 7, 7, 2, 2], 5, 30, 150 [8, 7, 6, 5, 0], 1, 120, 120 [8, 7, 6, 4, 1], 3, 120, 360 [8, 7, 6, 3, 2], 2, 120, 240 [8, 7, 5, 5, 1], 7, 60, 420 [8, 7, 5, 4, 2], 8, 120, 960 [8, 7, 5, 3, 3], 2, 60, 120 [8, 7, 4, 4, 3], 4, 60, 240 [8, 6, 6, 5, 1], 2, 60, 120 [8, 6, 6, 4, 2], 2, 60, 120 [8, 6, 5, 5, 2], 4, 60, 240 [8, 6, 5, 4, 3], 4, 120, 480 [8, 6, 4, 4, 4], 6, 20, 120 [8, 5, 5, 5, 3], 6, 20, 120 [8, 5, 5, 4, 4], 14, 30, 420 [7, 7, 7, 4, 1], 2, 20, 40 [7, 7, 7, 3, 2], 6, 20, 120 [7, 7, 6, 5, 1], 4, 60, 240 [7, 7, 6, 4, 2], 4, 60, 240 [7, 7, 5, 5, 2], 14, 30, 420 [7, 7, 5, 4, 3], 8, 60, 480 [7, 7, 4, 4, 4], 12, 10, 120 [7, 6, 6, 5, 2], 4, 60, 240 [7, 6, 5, 5, 3], 6, 60, 360 [7, 6, 5, 4, 4], 8, 60, 480 [7, 5, 5, 5, 4], 21, 20, 420 [6, 6, 5, 5, 4], 6, 30, 180 [6, 5, 5, 5, 5], 12, 5, 60 [12, 9, 2, 2, 2], 1, 20, 20 [12, 7, 5, 2, 1], 1, 120, 120 [12, 7, 4, 3, 1], 1, 120, 120 [12, 7, 4, 2, 2], 2, 60, 120 [12, 6, 5, 2, 2], 1, 60, 60 [12, 5, 5, 5, 0], 1, 20, 20 [12, 5, 5, 4, 1], 2, 60, 120 [12, 5, 5, 3, 2], 1, 60, 60 [12, 5, 4, 4, 2], 2, 60, 120 [12, 4, 4, 4, 3], 2, 20, 40 [10, 10, 4, 2, 1], 1, 60, 60 [10, 9, 5, 2, 1], 1, 120, 120 [10, 9, 4, 3, 1], 1, 120, 120 [10, 9, 4, 2, 2], 2, 60, 120 [10, 8, 7, 1, 1], 1, 60, 60 [10, 8, 6, 2, 1], 1, 120, 120 [10, 8, 5, 4, 0], 1, 120, 120 [10, 8, 5, 3, 1], 1, 120, 120 [10, 8, 4, 4, 1], 3, 60, 180 [10, 8, 4, 3, 2], 1, 120, 120 [10, 7, 7, 2, 1], 3, 60, 180 [10, 7, 6, 4, 0], 1, 120, 120 [10, 7, 6, 3, 1], 2, 120, 240 [10, 7, 6, 2, 2], 2, 60, 120 [10, 7, 5, 5, 0], 2, 60, 120 [10, 7, 5, 4, 1], 6, 120, 720 [10, 7, 5, 3, 2], 2, 120, 240 [10, 7, 4, 4, 2], 7, 60, 420 [10, 7, 4, 3, 3], 2, 60, 120 [10, 6, 6, 4, 1], 2, 60, 120 [10, 6, 5, 5, 1], 2, 60, 120 [10, 6, 5, 4, 2], 3, 120, 360 [10, 6, 4, 4, 3], 4, 60, 240 [10, 5, 5, 4, 3], 4, 60, 240 [10, 5, 4, 4, 4], 12, 20, 240 [9, 9, 5, 2, 2], 2, 30, 60 [9, 8, 7, 2, 1], 1, 120, 120 [9, 8, 6, 2, 2], 1, 60, 60 [9, 8, 5, 5, 0], 1, 60, 60 [9, 8, 5, 4, 1], 2, 120, 240 [9, 8, 5, 3, 2], 1, 120, 120 [9, 8, 4, 4, 2], 2, 60, 120 [9, 7, 7, 3, 1], 2, 60, 120 [9, 7, 7, 2, 2], 4, 30, 120 [9, 7, 6, 4, 1], 1, 120, 120 [9, 7, 5, 5, 1], 4, 60, 240 [9, 7, 5, 4, 2], 5, 120, 600 [9, 7, 4, 4, 3], 4, 60, 240 [9, 6, 5, 5, 2], 3, 60, 180 [9, 6, 4, 4, 4], 2, 20, 40 [9, 5, 5, 5, 3], 4, 20, 80 [9, 5, 5, 4, 4], 8, 30, 240 [8, 8, 8, 3, 0], 1, 20, 20 [8, 8, 7, 4, 0], 2, 60, 120 [8, 8, 7, 3, 1], 2, 60, 120 [8, 8, 6, 5, 0], 1, 60, 60 [8, 8, 6, 4, 1], 2, 60, 120 [8, 8, 6, 3, 2], 1, 60, 60 [8, 8, 5, 3, 3], 2, 30, 60 [8, 8, 4, 4, 3], 4, 30, 120 [8, 7, 7, 5, 0], 2, 60, 120 [8, 7, 7, 4, 1], 7, 60, 420 [8, 7, 7, 3, 2], 2, 60, 120 [8, 7, 6, 5, 1], 3, 120, 360 [8, 7, 6, 4, 2], 3, 120, 360 [8, 7, 5, 4, 3], 5, 120, 600 [8, 7, 4, 4, 4], 15, 20, 300 [8, 6, 6, 5, 2], 2, 60, 120 [8, 6, 5, 5, 3], 3, 60, 180 [8, 6, 5, 4, 4], 6, 60, 360 [7, 7, 7, 6, 0], 2, 20, 40 [7, 7, 7, 5, 1], 12, 20, 240 [7, 7, 7, 4, 2], 15, 20, 300 [7, 7, 7, 3, 3], 4, 10, 40 [7, 7, 6, 6, 1], 4, 30, 120 [7, 7, 6, 5, 2], 6, 60, 360 [7, 7, 6, 4, 3], 8, 60, 480 [7, 7, 5, 5, 3], 8, 30, 240 [7, 7, 5, 4, 4], 24, 30, 720 [7, 6, 6, 4, 4], 8, 30, 240 [7, 6, 5, 5, 4], 10, 60, 600 [6, 6, 5, 5, 5], 6, 10, 60 [12, 9, 4, 2, 1], 1, 120, 120 [12, 7, 7, 1, 1], 1, 30, 30 [12, 7, 6, 2, 1], 1, 120, 120 [12, 7, 5, 4, 0], 1, 120, 120 [12, 7, 5, 3, 1], 1, 120, 120 [12, 7, 4, 4, 1], 3, 60, 180 [12, 7, 4, 3, 2], 1, 120, 120 [12, 6, 5, 4, 1], 1, 120, 120 [12, 6, 4, 4, 2], 2, 60, 120 [12, 5, 4, 4, 3], 2, 60, 120 [12, 4, 4, 4, 4], 6, 5, 30 [10, 10, 6, 1, 1], 1, 30, 30 [10, 10, 4, 4, 0], 1, 30, 30 [10, 10, 4, 3, 1], 1, 60, 60 [10, 9, 7, 1, 1], 1, 60, 60 [10, 9, 6, 2, 1], 1, 120, 120 [10, 9, 5, 4, 0], 1, 120, 120 [10, 9, 5, 3, 1], 1, 120, 120 [10, 9, 4, 4, 1], 3, 60, 180 [10, 9, 4, 3, 2], 1, 120, 120 [10, 8, 7, 3, 0], 1, 120, 120 [10, 8, 6, 4, 0], 1, 120, 120 [10, 8, 6, 3, 1], 1, 120, 120 [10, 8, 4, 3, 3], 2, 60, 120 [10, 7, 7, 4, 0], 3, 60, 180 [10, 7, 7, 3, 1], 3, 60, 180 [10, 7, 6, 5, 0], 1, 120, 120 [10, 7, 6, 4, 1], 4, 120, 480 [10, 7, 6, 3, 2], 1, 120, 120 [10, 7, 5, 3, 3], 2, 60, 120 [10, 7, 4, 4, 3], 7, 60, 420 [10, 6, 6, 5, 1], 2, 60, 120 [10, 6, 6, 4, 2], 2, 60, 120 [10, 6, 5, 4, 3], 3, 120, 360 [10, 6, 4, 4, 4], 9, 20, 180 [9, 9, 7, 2, 1], 2, 60, 120 [9, 9, 5, 4, 1], 2, 60, 120 [9, 9, 4, 4, 2], 4, 30, 120 [9, 8, 7, 4, 0], 1, 120, 120 [9, 8, 7, 3, 1], 1, 120, 120 [9, 8, 6, 4, 1], 1, 120, 120 [9, 8, 4, 4, 3], 2, 60, 120 [9, 7, 7, 5, 0], 2, 60, 120 [9, 7, 7, 4, 1], 7, 60, 420 [9, 7, 7, 3, 2], 2, 60, 120 [9, 7, 6, 5, 1], 3, 120, 360 [9, 7, 6, 4, 2], 4, 120, 480 [9, 7, 5, 4, 3], 5, 120, 600 [9, 7, 4, 4, 4], 15, 20, 300 [9, 6, 5, 4, 4], 6, 60, 360 [8, 7, 7, 6, 0], 2, 60, 120 [8, 7, 7, 3, 3], 4, 30, 120 [8, 7, 6, 6, 1], 2, 60, 120 [8, 7, 6, 4, 3], 4, 120, 480 [8, 6, 6, 4, 4], 4, 30, 120 [7, 7, 7, 7, 0], 6, 5, 30 [7, 7, 7, 6, 1], 9, 20, 180 [7, 7, 7, 4, 3], 15, 20, 300 [7, 7, 6, 6, 2], 4, 30, 120 [7, 7, 6, 5, 3], 6, 60, 360 [7, 7, 6, 4, 4], 19, 30, 570 [7, 6, 6, 5, 4], 8, 60, 480 [12, 9, 6, 1, 1], 1, 60, 60 [12, 9, 4, 4, 0], 1, 60, 60 [12, 9, 4, 3, 1], 1, 120, 120 [12, 7, 7, 3, 0], 1, 60, 60 [12, 7, 6, 4, 0], 1, 120, 120 [12, 7, 6, 3, 1], 1, 120, 120 [12, 7, 4, 3, 3], 2, 60, 120 [12, 6, 6, 4, 1], 2, 60, 120 [12, 6, 4, 4, 3], 3, 60, 180 [10, 10, 6, 3, 0], 1, 60, 60 [10, 10, 3, 3, 3], 2, 10, 20 [10, 9, 7, 3, 0], 1, 120, 120 [10, 9, 6, 4, 0], 1, 120, 120 [10, 9, 6, 3, 1], 1, 120, 120 [10, 9, 4, 3, 3], 2, 60, 120 [10, 7, 6, 6, 0], 2, 60, 120 [10, 7, 6, 3, 3], 4, 60, 240 [10, 6, 6, 6, 1], 2, 20, 40 [10, 6, 6, 4, 3], 4, 60, 240 [9, 9, 9, 1, 1], 2, 10, 20 [9, 9, 7, 4, 0], 2, 60, 120 [9, 9, 7, 3, 1], 2, 60, 120 [9, 9, 6, 4, 1], 4, 60, 240 [9, 9, 4, 4, 3], 6, 30, 180 [9, 7, 7, 6, 0], 3, 60, 180 [9, 7, 7, 3, 3], 6, 30, 180 [9, 7, 6, 6, 1], 4, 60, 240 [9, 7, 6, 4, 3], 7, 120, 840 [9, 6, 6, 4, 4], 10, 30, 300 [7, 7, 6, 6, 3], 10, 30, 300 [7, 6, 6, 6, 4], 12, 20, 240 [12, 9, 6, 3, 0], 1, 120, 120 [12, 9, 3, 3, 3], 2, 20, 40 [12, 6, 6, 6, 0], 2, 20, 40 [12, 6, 6, 3, 3], 4, 30, 120 [9, 9, 9, 3, 0], 2, 20, 40 [9, 9, 6, 6, 0], 4, 30, 120 [9, 9, 6, 3, 3], 8, 30, 240 [9, 6, 6, 6, 3], 14, 20, 280 [6, 6, 6, 6, 6], 24, 1, 24 Sum of numbers in last column=, 59049 The most probable vector is, [8, 7, 5, 4, 2], which arises in, 960, out of, 3, to the power, 10 that equals, 59049, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 5, players The sole champion score-vector is, [8, 7, 5, 4, 2] the corresponding record is, 960 ----------------------------------------------------------------------------\ ----- 6, Players. Symmetric Function:, 6 / 6 \ --------' | --------' | ' | | |' | | 3 3 | | | | | | (a[i] + a[j] + 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, 5, 5, 5, 5, 5], 1, 1, 1 [7, 5, 5, 5, 5, 4], 1, 30, 30 [9, 5, 5, 5, 4, 4], 1, 60, 60 [7, 7, 5, 5, 5, 3], 1, 60, 60 [7, 7, 5, 5, 4, 4], 2, 90, 180 [7, 6, 5, 5, 5, 4], 1, 120, 120 [11, 5, 5, 4, 4, 4], 1, 60, 60 [9, 7, 5, 5, 4, 3], 1, 360, 360 [9, 7, 5, 4, 4, 4], 3, 120, 360 [9, 6, 5, 5, 5, 3], 1, 120, 120 [9, 6, 5, 5, 4, 4], 2, 180, 360 [8, 7, 5, 5, 4, 4], 1, 180, 180 [7, 7, 7, 5, 5, 2], 1, 60, 60 [7, 7, 7, 5, 4, 3], 3, 120, 360 [7, 7, 7, 4, 4, 4], 6, 20, 120 [7, 7, 6, 5, 5, 3], 2, 180, 360 [7, 7, 6, 5, 4, 4], 4, 180, 720 [7, 7, 5, 5, 5, 4], 3, 60, 180 [7, 6, 6, 5, 5, 4], 2, 180, 360 [6, 6, 6, 5, 5, 5], 2, 20, 40 [13, 5, 4, 4, 4, 4], 1, 30, 30 [11, 7, 5, 4, 4, 3], 1, 360, 360 [11, 7, 4, 4, 4, 4], 4, 30, 120 [11, 6, 5, 5, 4, 3], 1, 360, 360 [11, 6, 5, 4, 4, 4], 3, 120, 360 [10, 7, 5, 4, 4, 4], 1, 120, 120 [9, 9, 5, 5, 3, 3], 1, 90, 90 [9, 9, 5, 4, 4, 3], 2, 180, 360 [9, 9, 4, 4, 4, 4], 6, 15, 90 [9, 8, 5, 5, 4, 3], 1, 360, 360 [9, 8, 5, 4, 4, 4], 3, 120, 360 [9, 7, 7, 5, 4, 2], 1, 360, 360 [9, 7, 7, 5, 3, 3], 2, 180, 360 [9, 7, 7, 4, 4, 3], 5, 180, 900 [9, 7, 6, 5, 5, 2], 1, 360, 360 [9, 7, 6, 5, 4, 3], 4, 720, 2880 [9, 7, 6, 4, 4, 4], 9, 120, 1080 [9, 7, 5, 5, 5, 3], 3, 120, 360 [9, 7, 5, 5, 4, 4], 4, 180, 720 [9, 6, 6, 5, 5, 3], 3, 180, 540 [9, 6, 6, 5, 4, 4], 6, 180, 1080 [9, 6, 5, 5, 5, 4], 3, 120, 360 [8, 7, 7, 5, 4, 3], 2, 360, 720 [8, 7, 7, 4, 4, 4], 6, 60, 360 [8, 7, 6, 5, 5, 3], 1, 360, 360 [8, 7, 6, 5, 4, 4], 3, 360, 1080 [8, 6, 6, 5, 5, 4], 2, 180, 360 [7, 7, 7, 7, 5, 1], 1, 30, 30 [7, 7, 7, 7, 4, 2], 4, 30, 120 [7, 7, 7, 7, 3, 3], 6, 15, 90 [7, 7, 7, 6, 5, 2], 3, 120, 360 [7, 7, 7, 6, 4, 3], 9, 120, 1080 [7, 7, 7, 5, 5, 3], 6, 60, 360 [7, 7, 7, 5, 4, 4], 11, 60, 660 [7, 7, 6, 6, 5, 3], 6, 180, 1080 [7, 7, 6, 6, 4, 4], 12, 90, 1080 [7, 7, 6, 5, 5, 4], 6, 180, 1080 [7, 6, 6, 6, 5, 4], 8, 120, 960 [7, 6, 6, 5, 5, 5], 6, 60, 360 [6, 6, 6, 6, 5, 5], 6, 15, 90 [15, 4, 4, 4, 4, 4], 1, 6, 6 [13, 7, 4, 4, 4, 3], 1, 120, 120 [13, 6, 5, 4, 4, 3], 1, 360, 360 [13, 6, 4, 4, 4, 4], 4, 30, 120 [12, 7, 4, 4, 4, 4], 1, 30, 30 [11, 9, 5, 4, 3, 3], 1, 360, 360 [11, 9, 4, 4, 4, 3], 3, 120, 360 [11, 8, 5, 5, 3, 3], 1, 180, 180 [11, 8, 5, 4, 4, 3], 2, 360, 720 [11, 8, 4, 4, 4, 4], 6, 30, 180 [11, 7, 7, 4, 4, 2], 1, 180, 180 [11, 7, 7, 4, 3, 3], 2, 180, 360 [11, 7, 6, 5, 4, 2], 1, 720, 720 [11, 7, 6, 5, 3, 3], 2, 360, 720 [11, 7, 6, 4, 4, 3], 6, 360, 2160 [11, 7, 5, 5, 4, 3], 2, 360, 720 [11, 7, 5, 4, 4, 4], 3, 120, 360 [11, 6, 6, 5, 5, 2], 1, 180, 180 [11, 6, 6, 5, 4, 3], 5, 360, 1800 [11, 6, 6, 4, 4, 4], 12, 60, 720 [11, 6, 5, 5, 5, 3], 3, 120, 360 [11, 6, 5, 5, 4, 4], 4, 180, 720 [10, 9, 5, 4, 4, 3], 1, 360, 360 [10, 9, 4, 4, 4, 4], 4, 30, 120 [10, 7, 7, 4, 4, 3], 2, 180, 360 [10, 7, 6, 5, 4, 3], 1, 720, 720 [10, 7, 6, 4, 4, 4], 4, 120, 480 [10, 6, 6, 5, 4, 4], 2, 180, 360 [9, 9, 7, 5, 3, 2], 1, 360, 360 [9, 9, 7, 4, 4, 2], 2, 180, 360 [9, 9, 7, 4, 3, 3], 5, 180, 900 [9, 9, 6, 5, 4, 2], 2, 360, 720 [9, 9, 6, 5, 3, 3], 4, 180, 720 [9, 9, 6, 4, 4, 3], 10, 180, 1800 [9, 9, 5, 5, 5, 2], 3, 60, 180 [9, 9, 5, 5, 4, 3], 6, 180, 1080 [9, 9, 5, 4, 4, 4], 6, 60, 360 [9, 8, 7, 5, 4, 2], 1, 720, 720 [9, 8, 7, 5, 3, 3], 3, 360, 1080 [9, 8, 7, 4, 4, 3], 7, 360, 2520 [9, 8, 6, 5, 5, 2], 1, 360, 360 [9, 8, 6, 5, 4, 3], 5, 720, 3600 [9, 8, 6, 4, 4, 4], 12, 120, 1440 [9, 8, 5, 5, 5, 3], 3, 120, 360 [9, 8, 5, 5, 4, 4], 4, 180, 720 [9, 7, 7, 7, 4, 1], 1, 120, 120 [9, 7, 7, 7, 3, 2], 3, 120, 360 [9, 7, 7, 6, 5, 1], 1, 360, 360 [9, 7, 7, 6, 4, 2], 6, 360, 2160 [9, 7, 7, 6, 3, 3], 10, 180, 1800 [9, 7, 7, 5, 5, 2], 4, 180, 720 [9, 7, 7, 5, 4, 3], 10, 360, 3600 [9, 7, 7, 4, 4, 4], 13, 60, 780 [9, 7, 6, 6, 5, 2], 5, 360, 1800 [9, 7, 6, 6, 4, 3], 17, 360, 6120 [9, 7, 6, 5, 5, 3], 10, 360, 3600 [9, 7, 6, 5, 4, 4], 13, 360, 4680 [9, 7, 5, 5, 5, 4], 6, 120, 720 [9, 6, 6, 6, 5, 3], 12, 120, 1440 [9, 6, 6, 6, 4, 4], 26, 60, 1560 [9, 6, 6, 5, 5, 4], 14, 180, 2520 [9, 6, 5, 5, 5, 5], 12, 30, 360 [8, 8, 7, 5, 4, 3], 2, 360, 720 [8, 8, 7, 4, 4, 4], 6, 60, 360 [8, 8, 6, 5, 5, 3], 2, 180, 360 [8, 8, 6, 5, 4, 4], 4, 180, 720 [8, 7, 7, 7, 4, 2], 3, 120, 360 [8, 7, 7, 7, 3, 3], 6, 60, 360 [8, 7, 7, 6, 5, 2], 2, 360, 720 [8, 7, 7, 6, 4, 3], 10, 360, 3600 [8, 7, 7, 5, 5, 3], 4, 180, 720 [8, 7, 7, 5, 4, 4], 5, 180, 900 [8, 7, 6, 6, 5, 3], 7, 360, 2520 [8, 7, 6, 6, 4, 4], 16, 180, 2880 [8, 7, 6, 5, 5, 4], 6, 360, 2160 [8, 6, 6, 6, 5, 4], 12, 120, 1440 [8, 6, 6, 5, 5, 5], 9, 60, 540 [7, 7, 7, 7, 7, 0], 1, 6, 6 [7, 7, 7, 7, 6, 1], 4, 30, 120 [7, 7, 7, 7, 5, 2], 6, 30, 180 [7, 7, 7, 7, 4, 3], 17, 30, 510 [7, 7, 7, 6, 6, 2], 12, 60, 720 [7, 7, 7, 6, 5, 3], 15, 120, 1800 [7, 7, 7, 6, 4, 4], 21, 60, 1260 [7, 7, 7, 5, 5, 4], 6, 60, 360 [7, 7, 6, 6, 6, 3], 26, 60, 1560 [7, 7, 6, 6, 5, 4], 22, 180, 3960 [7, 7, 6, 5, 5, 5], 12, 60, 720 [7, 6, 6, 6, 6, 4], 38, 30, 1140 [7, 6, 6, 6, 5, 5], 24, 60, 1440 [6, 6, 6, 6, 6, 5], 24, 6, 144 [15, 6, 4, 4, 4, 3], 1, 120, 120 [13, 9, 4, 4, 3, 3], 1, 180, 180 [13, 8, 5, 4, 3, 3], 1, 360, 360 [13, 8, 4, 4, 4, 3], 3, 120, 360 [13, 7, 6, 4, 4, 2], 1, 360, 360 [13, 7, 6, 4, 3, 3], 2, 360, 720 [13, 7, 5, 4, 4, 3], 1, 360, 360 [13, 6, 6, 5, 4, 2], 1, 360, 360 [13, 6, 6, 5, 3, 3], 2, 180, 360 [13, 6, 6, 4, 4, 3], 7, 180, 1260 [13, 6, 5, 5, 4, 3], 2, 360, 720 [13, 6, 5, 4, 4, 4], 3, 120, 360 [12, 9, 4, 4, 4, 3], 1, 120, 120 [12, 7, 6, 4, 4, 3], 1, 360, 360 [12, 6, 6, 4, 4, 4], 2, 60, 120 [11, 11, 5, 3, 3, 3], 1, 60, 60 [11, 11, 4, 4, 3, 3], 2, 90, 180 [11, 10, 5, 4, 3, 3], 1, 360, 360 [11, 10, 4, 4, 4, 3], 3, 120, 360 [11, 9, 7, 4, 3, 2], 1, 720, 720 [11, 9, 7, 3, 3, 3], 3, 120, 360 [11, 9, 6, 5, 3, 2], 1, 720, 720 [11, 9, 6, 4, 4, 2], 3, 360, 1080 [11, 9, 6, 4, 3, 3], 7, 360, 2520 [11, 9, 5, 5, 4, 2], 2, 360, 720 [11, 9, 5, 5, 3, 3], 4, 180, 720 [11, 9, 5, 4, 4, 3], 5, 360, 1800 [11, 8, 7, 5, 3, 2], 1, 720, 720 [11, 8, 7, 4, 4, 2], 2, 360, 720 [11, 8, 7, 4, 3, 3], 6, 360, 2160 [11, 8, 6, 5, 4, 2], 2, 720, 1440 [11, 8, 6, 5, 3, 3], 5, 360, 1800 [11, 8, 6, 4, 4, 3], 12, 360, 4320 [11, 8, 5, 5, 5, 2], 3, 120, 360 [11, 8, 5, 5, 4, 3], 6, 360, 2160 [11, 8, 5, 4, 4, 4], 6, 120, 720 [11, 7, 7, 6, 4, 1], 1, 360, 360 [11, 7, 7, 6, 3, 2], 3, 360, 1080 [11, 7, 7, 5, 4, 2], 2, 360, 720 [11, 7, 7, 5, 3, 3], 6, 180, 1080 [11, 7, 7, 4, 4, 3], 6, 180, 1080 [11, 7, 6, 6, 5, 1], 1, 360, 360 [11, 7, 6, 6, 4, 2], 8, 360, 2880 [11, 7, 6, 6, 3, 3], 14, 180, 2520 [11, 7, 6, 5, 5, 2], 4, 360, 1440 [11, 7, 6, 5, 4, 3], 11, 720, 7920 [11, 7, 6, 4, 4, 4], 9, 120, 1080 [11, 7, 5, 5, 5, 3], 6, 120, 720 [11, 7, 5, 5, 4, 4], 4, 180, 720 [11, 6, 6, 6, 5, 2], 7, 120, 840 [11, 6, 6, 6, 4, 3], 27, 120, 3240 [11, 6, 6, 5, 5, 3], 14, 180, 2520 [11, 6, 6, 5, 4, 4], 18, 180, 3240 [11, 6, 5, 5, 5, 4], 12, 120, 1440 [11, 5, 5, 5, 5, 5], 20, 6, 120 [10, 9, 7, 4, 4, 2], 1, 360, 360 [10, 9, 7, 4, 3, 3], 3, 360, 1080 [10, 9, 6, 5, 4, 2], 1, 720, 720 [10, 9, 6, 5, 3, 3], 2, 360, 720 [10, 9, 6, 4, 4, 3], 7, 360, 2520 [10, 9, 5, 5, 4, 3], 2, 360, 720 [10, 9, 5, 4, 4, 4], 3, 120, 360 [10, 8, 7, 5, 3, 3], 1, 360, 360 [10, 8, 7, 4, 4, 3], 3, 360, 1080 [10, 8, 6, 5, 4, 3], 2, 720, 1440 [10, 8, 6, 4, 4, 4], 6, 120, 720 [10, 7, 7, 6, 4, 2], 2, 360, 720 [10, 7, 7, 6, 3, 3], 4, 180, 720 [10, 7, 7, 5, 4, 3], 2, 360, 720 [10, 7, 6, 6, 5, 2], 1, 360, 360 [10, 7, 6, 6, 4, 3], 9, 360, 3240 [10, 7, 6, 5, 5, 3], 2, 360, 720 [10, 7, 6, 5, 4, 4], 4, 360, 1440 [10, 6, 6, 6, 5, 3], 6, 120, 720 [10, 6, 6, 6, 4, 4], 18, 60, 1080 [10, 6, 6, 5, 5, 4], 6, 180, 1080 [9, 9, 9, 5, 2, 2], 1, 60, 60 [9, 9, 9, 4, 3, 2], 3, 120, 360 [9, 9, 9, 3, 3, 3], 6, 20, 120 [9, 9, 8, 5, 3, 2], 2, 360, 720 [9, 9, 8, 4, 4, 2], 4, 180, 720 [9, 9, 8, 4, 3, 3], 10, 180, 1800 [9, 9, 7, 7, 3, 1], 1, 180, 180 [9, 9, 7, 7, 2, 2], 2, 90, 180 [9, 9, 7, 6, 4, 1], 2, 360, 720 [9, 9, 7, 6, 3, 2], 7, 360, 2520 [9, 9, 7, 5, 5, 1], 2, 180, 360 [9, 9, 7, 5, 4, 2], 8, 360, 2880 [9, 9, 7, 5, 3, 3], 12, 180, 2160 [9, 9, 7, 4, 4, 3], 13, 180, 2340 [9, 9, 6, 6, 5, 1], 2, 180, 360 [9, 9, 6, 6, 4, 2], 14, 180, 2520 [9, 9, 6, 6, 3, 3], 24, 90, 2160 [9, 9, 6, 5, 5, 2], 10, 180, 1800 [9, 9, 6, 5, 4, 3], 19, 360, 6840 [9, 9, 6, 4, 4, 4], 14, 60, 840 [9, 9, 5, 5, 5, 3], 15, 60, 900 [9, 9, 5, 5, 4, 4], 14, 90, 1260 [9, 8, 8, 5, 4, 2], 2, 360, 720 [9, 8, 8, 5, 3, 3], 5, 180, 900 [9, 8, 8, 4, 4, 3], 12, 180, 2160 [9, 8, 7, 7, 4, 1], 1, 360, 360 [9, 8, 7, 7, 3, 2], 5, 360, 1800 [9, 8, 7, 6, 5, 1], 1, 720, 720 [9, 8, 7, 6, 4, 2], 9, 720, 6480 [9, 8, 7, 6, 3, 3], 17, 360, 6120 [9, 8, 7, 5, 5, 2], 6, 360, 2160 [9, 8, 7, 5, 4, 3], 13, 720, 9360 [9, 8, 7, 4, 4, 4], 9, 120, 1080 [9, 8, 6, 6, 5, 2], 7, 360, 2520 [9, 8, 6, 6, 4, 3], 29, 360, 10440 [9, 8, 6, 5, 5, 3], 16, 360, 5760 [9, 8, 6, 5, 4, 4], 19, 360, 6840 [9, 8, 5, 5, 5, 4], 12, 120, 1440 [9, 7, 7, 7, 6, 0], 1, 120, 120 [9, 7, 7, 7, 5, 1], 3, 120, 360 [9, 7, 7, 7, 4, 2], 9, 120, 1080 [9, 7, 7, 7, 3, 3], 17, 60, 1020 [9, 7, 7, 6, 6, 1], 7, 180, 1260 [9, 7, 7, 6, 5, 2], 14, 360, 5040 [9, 7, 7, 6, 4, 3], 24, 360, 8640 [9, 7, 7, 5, 5, 3], 16, 180, 2880 [9, 7, 7, 5, 4, 4], 12, 180, 2160 [9, 7, 6, 6, 6, 2], 27, 120, 3240 [9, 7, 6, 6, 5, 3], 33, 360, 11880 [9, 7, 6, 6, 4, 4], 33, 180, 5940 [9, 7, 6, 5, 5, 4], 26, 360, 9360 [9, 7, 5, 5, 5, 5], 24, 30, 720 [9, 6, 6, 6, 6, 3], 68, 30, 2040 [9, 6, 6, 6, 5, 4], 51, 120, 6120 [9, 6, 6, 5, 5, 5], 42, 60, 2520 [8, 8, 8, 5, 5, 2], 2, 60, 120 [8, 8, 8, 5, 4, 3], 6, 120, 720 [8, 8, 8, 4, 4, 4], 12, 20, 240 [8, 8, 7, 7, 4, 2], 4, 180, 720 [8, 8, 7, 7, 3, 3], 10, 90, 900 [8, 8, 7, 6, 5, 2], 4, 360, 1440 [8, 8, 7, 6, 4, 3], 18, 360, 6480 [8, 8, 7, 5, 5, 3], 8, 180, 1440 [8, 8, 7, 5, 4, 4], 8, 180, 1440 [8, 8, 6, 6, 5, 3], 14, 180, 2520 [8, 8, 6, 6, 4, 4], 32, 90, 2880 [8, 8, 6, 5, 5, 4], 16, 180, 2880 [8, 8, 5, 5, 5, 5], 24, 15, 360 [8, 7, 7, 7, 6, 1], 3, 120, 360 [8, 7, 7, 7, 5, 2], 6, 120, 720 [8, 7, 7, 7, 4, 3], 12, 120, 1440 [8, 7, 7, 6, 6, 2], 16, 180, 2880 [8, 7, 7, 6, 5, 3], 21, 360, 7560 [8, 7, 7, 6, 4, 4], 18, 180, 3240 [8, 7, 7, 5, 5, 4], 8, 180, 1440 [8, 7, 6, 6, 6, 3], 45, 120, 5400 [8, 7, 6, 6, 5, 4], 34, 360, 12240 [8, 7, 6, 5, 5, 5], 24, 120, 2880 [8, 6, 6, 6, 6, 4], 72, 30, 2160 [8, 6, 6, 6, 5, 5], 42, 60, 2520 [7, 7, 7, 7, 6, 2], 12, 30, 360 [7, 7, 7, 7, 5, 3], 12, 30, 360 [7, 7, 7, 6, 6, 3], 37, 60, 2220 [7, 7, 7, 6, 5, 4], 21, 120, 2520 [7, 7, 7, 5, 5, 5], 12, 20, 240 [7, 7, 6, 6, 6, 4], 60, 60, 3600 [7, 7, 6, 6, 5, 5], 44, 90, 3960 [7, 6, 6, 6, 6, 5], 84, 30, 2520 [6, 6, 6, 6, 6, 6], 160, 1, 160 [15, 8, 4, 4, 3, 3], 1, 180, 180 [15, 6, 6, 4, 4, 2], 1, 180, 180 [15, 6, 6, 4, 3, 3], 2, 180, 360 [15, 6, 5, 4, 4, 3], 1, 360, 360 [13, 11, 4, 3, 3, 3], 1, 120, 120 [13, 10, 5, 3, 3, 3], 1, 120, 120 [13, 10, 4, 4, 3, 3], 2, 180, 360 [13, 9, 6, 4, 3, 2], 1, 720, 720 [13, 9, 6, 3, 3, 3], 3, 120, 360 [13, 9, 5, 4, 4, 2], 1, 360, 360 [13, 9, 5, 4, 3, 3], 2, 360, 720 [13, 8, 7, 4, 3, 2], 1, 720, 720 [13, 8, 7, 3, 3, 3], 3, 120, 360 [13, 8, 6, 5, 3, 2], 1, 720, 720 [13, 8, 6, 4, 4, 2], 3, 360, 1080 [13, 8, 6, 4, 3, 3], 8, 360, 2880 [13, 8, 5, 5, 4, 2], 2, 360, 720 [13, 8, 5, 5, 3, 3], 4, 180, 720 [13, 8, 5, 4, 4, 3], 5, 360, 1800 [13, 7, 7, 4, 3, 3], 2, 180, 360 [13, 7, 6, 6, 4, 1], 1, 360, 360 [13, 7, 6, 6, 3, 2], 3, 360, 1080 [13, 7, 6, 5, 4, 2], 2, 720, 1440 [13, 7, 6, 5, 3, 3], 5, 360, 1800 [13, 7, 6, 4, 4, 3], 4, 360, 1440 [13, 7, 5, 5, 4, 3], 2, 360, 720 [13, 6, 6, 6, 5, 1], 1, 120, 120 [13, 6, 6, 6, 4, 2], 10, 120, 1200 [13, 6, 6, 6, 3, 3], 18, 60, 1080 [13, 6, 6, 5, 5, 2], 4, 180, 720 [13, 6, 6, 5, 4, 3], 12, 360, 4320 [13, 6, 6, 4, 4, 4], 6, 60, 360 [13, 6, 5, 5, 5, 3], 6, 120, 720 [13, 6, 5, 5, 4, 4], 6, 180, 1080 [13, 5, 5, 5, 5, 4], 8, 30, 240 [12, 11, 4, 4, 3, 3], 1, 180, 180 [12, 9, 6, 4, 4, 2], 1, 360, 360 [12, 9, 6, 4, 3, 3], 2, 360, 720 [12, 9, 5, 4, 4, 3], 1, 360, 360 [12, 8, 7, 4, 3, 3], 1, 360, 360 [12, 8, 6, 4, 4, 3], 2, 360, 720 [12, 7, 6, 6, 4, 2], 1, 360, 360 [12, 7, 6, 6, 3, 3], 2, 180, 360 [12, 7, 6, 5, 4, 3], 1, 720, 720 [12, 6, 6, 6, 4, 3], 6, 120, 720 [12, 6, 6, 5, 4, 4], 3, 180, 540 [11, 11, 7, 3, 3, 2], 1, 180, 180 [11, 11, 6, 4, 3, 2], 2, 360, 720 [11, 11, 6, 3, 3, 3], 6, 60, 360 [11, 11, 5, 5, 3, 2], 2, 180, 360 [11, 11, 5, 4, 4, 2], 2, 180, 360 [11, 11, 5, 4, 3, 3], 5, 180, 900 [11, 10, 7, 4, 3, 2], 1, 720, 720 [11, 10, 7, 3, 3, 3], 4, 120, 480 [11, 10, 6, 5, 3, 2], 1, 720, 720 [11, 10, 6, 4, 4, 2], 3, 360, 1080 [11, 10, 6, 4, 3, 3], 8, 360, 2880 [11, 10, 5, 5, 4, 2], 2, 360, 720 [11, 10, 5, 5, 3, 3], 4, 180, 720 [11, 10, 5, 4, 4, 3], 5, 360, 1800 [11, 9, 9, 4, 2, 2], 1, 180, 180 [11, 9, 9, 3, 3, 2], 2, 180, 360 [11, 9, 8, 5, 2, 2], 1, 360, 360 [11, 9, 8, 4, 3, 2], 4, 720, 2880 [11, 9, 8, 3, 3, 3], 9, 120, 1080 [11, 9, 7, 6, 3, 1], 1, 720, 720 [11, 9, 7, 6, 2, 2], 2, 360, 720 [11, 9, 7, 5, 4, 1], 1, 720, 720 [11, 9, 7, 5, 3, 2], 5, 720, 3600 [11, 9, 7, 4, 4, 2], 4, 360, 1440 [11, 9, 7, 4, 3, 3], 8, 360, 2880 [11, 9, 6, 6, 4, 1], 3, 360, 1080 [11, 9, 6, 6, 3, 2], 11, 360, 3960 [11, 9, 6, 5, 5, 1], 2, 360, 720 [11, 9, 6, 5, 4, 2], 10, 720, 7200 [11, 9, 6, 5, 3, 3], 16, 360, 5760 [11, 9, 6, 4, 4, 3], 9, 360, 3240 [11, 9, 5, 5, 5, 2], 9, 120, 1080 [11, 9, 5, 5, 4, 3], 13, 360, 4680 [11, 9, 5, 4, 4, 4], 6, 120, 720 [11, 8, 8, 5, 3, 2], 3, 360, 1080 [11, 8, 8, 4, 4, 2], 6, 180, 1080 [11, 8, 8, 4, 3, 3], 15, 180, 2700 [11, 8, 7, 7, 3, 1], 1, 360, 360 [11, 8, 7, 7, 2, 2], 2, 180, 360 [11, 8, 7, 6, 4, 1], 2, 720, 1440 [11, 8, 7, 6, 3, 2], 9, 720, 6480 [11, 8, 7, 5, 5, 1], 2, 360, 720 [11, 8, 7, 5, 4, 2], 9, 720, 6480 [11, 8, 7, 5, 3, 3], 15, 360, 5400 [11, 8, 7, 4, 4, 3], 11, 360, 3960 [11, 8, 6, 6, 5, 1], 2, 360, 720 [11, 8, 6, 6, 4, 2], 18, 360, 6480 [11, 8, 6, 6, 3, 3], 34, 180, 6120 [11, 8, 6, 5, 5, 2], 12, 360, 4320 [11, 8, 6, 5, 4, 3], 25, 720, 18000 [11, 8, 6, 4, 4, 4], 12, 120, 1440 [11, 8, 5, 5, 5, 3], 21, 120, 2520 [11, 8, 5, 5, 4, 4], 18, 180, 3240 [11, 7, 7, 7, 3, 2], 6, 120, 720 [11, 7, 7, 6, 6, 0], 1, 180, 180 [11, 7, 7, 6, 5, 1], 3, 360, 1080 [11, 7, 7, 6, 4, 2], 9, 360, 3240 [11, 7, 7, 6, 3, 3], 19, 180, 3420 [11, 7, 7, 5, 5, 2], 10, 180, 1800 [11, 7, 7, 5, 4, 3], 11, 360, 3960 [11, 7, 6, 6, 6, 1], 10, 120, 1200 [11, 7, 6, 6, 5, 2], 20, 360, 7200 [11, 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1, 720, 720 [13, 12, 8, 7, 3, 0], 1, 720, 720 [13, 12, 8, 6, 4, 0], 1, 720, 720 [13, 12, 8, 6, 3, 1], 1, 720, 720 [13, 12, 8, 4, 3, 3], 2, 360, 720 [13, 12, 7, 7, 4, 0], 3, 360, 1080 [13, 12, 7, 7, 3, 1], 3, 360, 1080 [13, 12, 7, 6, 5, 0], 1, 720, 720 [13, 12, 7, 6, 4, 1], 4, 720, 2880 [13, 12, 7, 6, 3, 2], 1, 720, 720 [13, 12, 7, 5, 3, 3], 2, 360, 720 [13, 12, 7, 4, 4, 3], 7, 360, 2520 [13, 12, 6, 6, 5, 1], 2, 360, 720 [13, 12, 6, 6, 4, 2], 2, 360, 720 [13, 12, 6, 5, 4, 3], 3, 720, 2160 [13, 12, 6, 4, 4, 4], 9, 120, 1080 [13, 11, 10, 6, 3, 0], 1, 720, 720 [13, 11, 10, 3, 3, 3], 2, 120, 240 [13, 11, 9, 7, 3, 0], 1, 720, 720 [13, 11, 9, 6, 4, 0], 1, 720, 720 [13, 11, 9, 6, 3, 1], 1, 720, 720 [13, 11, 9, 4, 3, 3], 2, 360, 720 [13, 11, 7, 6, 6, 0], 2, 360, 720 [13, 11, 7, 6, 3, 3], 4, 360, 1440 [13, 11, 6, 6, 6, 1], 2, 120, 240 [13, 11, 6, 6, 4, 3], 4, 360, 1440 [13, 10, 10, 7, 3, 0], 3, 360, 1080 [13, 10, 10, 6, 4, 0], 3, 360, 1080 [13, 10, 10, 6, 3, 1], 3, 360, 1080 [13, 10, 10, 4, 3, 3], 6, 180, 1080 [13, 10, 9, 9, 1, 1], 2, 180, 360 [13, 10, 9, 8, 3, 0], 1, 720, 720 [13, 10, 9, 7, 4, 0], 4, 720, 2880 [13, 10, 9, 7, 3, 1], 4, 720, 2880 [13, 10, 9, 6, 5, 0], 1, 720, 720 [13, 10, 9, 6, 4, 1], 6, 720, 4320 [13, 10, 9, 6, 3, 2], 1, 720, 720 [13, 10, 9, 5, 3, 3], 2, 360, 720 [13, 10, 9, 4, 4, 3], 10, 360, 3600 [13, 10, 8, 6, 6, 0], 2, 360, 720 [13, 10, 8, 6, 3, 3], 4, 360, 1440 [13, 10, 7, 7, 6, 0], 7, 360, 2520 [13, 10, 7, 7, 3, 3], 14, 180, 2520 [13, 10, 7, 6, 6, 1], 8, 360, 2880 [13, 10, 7, 6, 4, 3], 15, 720, 10800 [13, 10, 6, 6, 6, 2], 2, 120, 240 [13, 10, 6, 6, 5, 3], 4, 360, 1440 [13, 10, 6, 6, 4, 4], 18, 180, 3240 [13, 9, 9, 9, 2, 1], 2, 120, 240 [13, 9, 9, 8, 4, 0], 2, 360, 720 [13, 9, 9, 8, 3, 1], 2, 360, 720 [13, 9, 9, 7, 5, 0], 2, 360, 720 [13, 9, 9, 7, 4, 1], 8, 360, 2880 [13, 9, 9, 7, 3, 2], 2, 360, 720 [13, 9, 9, 6, 5, 1], 4, 360, 1440 [13, 9, 9, 6, 4, 2], 4, 360, 1440 [13, 9, 9, 5, 4, 3], 6, 360, 2160 [13, 9, 9, 4, 4, 4], 18, 60, 1080 [13, 9, 8, 7, 6, 0], 3, 720, 2160 [13, 9, 8, 7, 3, 3], 6, 360, 2160 [13, 9, 8, 6, 6, 1], 4, 360, 1440 [13, 9, 8, 6, 4, 3], 7, 720, 5040 [13, 9, 7, 7, 7, 0], 9, 120, 1080 [13, 9, 7, 7, 6, 1], 11, 360, 3960 [13, 9, 7, 7, 4, 3], 20, 360, 7200 [13, 9, 7, 6, 6, 2], 4, 360, 1440 [13, 9, 7, 6, 5, 3], 7, 720, 5040 [13, 9, 7, 6, 4, 4], 24, 360, 8640 [13, 9, 6, 6, 5, 4], 10, 360, 3600 [13, 8, 7, 6, 6, 3], 10, 360, 3600 [13, 8, 6, 6, 6, 4], 12, 120, 1440 [13, 7, 7, 7, 6, 3], 30, 120, 3600 [13, 7, 7, 6, 6, 4], 34, 180, 6120 [13, 7, 6, 6, 6, 5], 12, 120, 1440 [12, 12, 12, 4, 2, 1], 2, 120, 240 [12, 12, 10, 7, 1, 1], 2, 180, 360 [12, 12, 10, 6, 2, 1], 2, 360, 720 [12, 12, 10, 5, 4, 0], 2, 360, 720 [12, 12, 10, 5, 3, 1], 2, 360, 720 [12, 12, 10, 4, 4, 1], 6, 180, 1080 [12, 12, 10, 4, 3, 2], 2, 360, 720 [12, 12, 9, 7, 2, 1], 4, 360, 1440 [12, 12, 9, 5, 4, 1], 4, 360, 1440 [12, 12, 9, 4, 4, 2], 8, 180, 1440 [12, 12, 8, 7, 4, 0], 2, 360, 720 [12, 12, 8, 7, 3, 1], 2, 360, 720 [12, 12, 8, 6, 4, 1], 2, 360, 720 [12, 12, 8, 4, 4, 3], 4, 180, 720 [12, 12, 7, 7, 5, 0], 4, 180, 720 [12, 12, 7, 7, 4, 1], 14, 180, 2520 [12, 12, 7, 7, 3, 2], 4, 180, 720 [12, 12, 7, 6, 5, 1], 6, 360, 2160 [12, 12, 7, 6, 4, 2], 8, 360, 2880 [12, 12, 7, 5, 4, 3], 10, 360, 3600 [12, 12, 7, 4, 4, 4], 30, 60, 1800 [12, 12, 6, 5, 4, 4], 12, 180, 2160 [12, 11, 10, 7, 3, 0], 1, 720, 720 [12, 11, 10, 6, 4, 0], 1, 720, 720 [12, 11, 10, 6, 3, 1], 1, 720, 720 [12, 11, 10, 4, 3, 3], 2, 360, 720 [12, 11, 9, 7, 4, 0], 1, 720, 720 [12, 11, 9, 7, 3, 1], 1, 720, 720 [12, 11, 9, 6, 4, 1], 1, 720, 720 [12, 11, 9, 4, 4, 3], 2, 360, 720 [12, 11, 7, 7, 6, 0], 2, 360, 720 [12, 11, 7, 7, 3, 3], 4, 180, 720 [12, 11, 7, 6, 6, 1], 2, 360, 720 [12, 11, 7, 6, 4, 3], 4, 720, 2880 [12, 11, 6, 6, 4, 4], 4, 180, 720 [12, 10, 10, 9, 1, 1], 3, 180, 540 [12, 10, 10, 8, 3, 0], 2, 360, 720 [12, 10, 10, 7, 4, 0], 7, 360, 2520 [12, 10, 10, 7, 3, 1], 7, 360, 2520 [12, 10, 10, 6, 5, 0], 2, 360, 720 [12, 10, 10, 6, 4, 1], 10, 360, 3600 [12, 10, 10, 6, 3, 2], 2, 360, 720 [12, 10, 10, 5, 3, 3], 4, 180, 720 [12, 10, 10, 4, 4, 3], 17, 180, 3060 [12, 10, 9, 9, 2, 1], 4, 360, 1440 [12, 10, 9, 8, 4, 0], 3, 720, 2160 [12, 10, 9, 8, 3, 1], 3, 720, 2160 [12, 10, 9, 7, 5, 0], 4, 720, 2880 [12, 10, 9, 7, 4, 1], 15, 720, 10800 [12, 10, 9, 7, 3, 2], 4, 720, 2880 [12, 10, 9, 6, 5, 1], 7, 720, 5040 [12, 10, 9, 6, 4, 2], 8, 720, 5760 [12, 10, 9, 5, 4, 3], 11, 720, 7920 [12, 10, 9, 4, 4, 4], 33, 120, 3960 [12, 10, 8, 7, 6, 0], 5, 720, 3600 [12, 10, 8, 7, 3, 3], 10, 360, 3600 [12, 10, 8, 6, 6, 1], 6, 360, 2160 [12, 10, 8, 6, 4, 3], 11, 720, 7920 [12, 10, 7, 7, 7, 0], 15, 120, 1800 [12, 10, 7, 7, 6, 1], 20, 360, 7200 [12, 10, 7, 7, 4, 3], 35, 360, 12600 [12, 10, 7, 6, 6, 2], 8, 360, 2880 [12, 10, 7, 6, 5, 3], 13, 720, 9360 [12, 10, 7, 6, 4, 4], 43, 360, 15480 [12, 10, 6, 6, 5, 4], 18, 360, 6480 [12, 9, 9, 8, 4, 1], 4, 360, 1440 [12, 9, 9, 7, 5, 1], 10, 360, 3600 [12, 9, 9, 7, 4, 2], 14, 360, 5040 [12, 9, 9, 5, 4, 4], 20, 180, 3600 [12, 9, 8, 7, 7, 0], 6, 360, 2160 [12, 9, 8, 7, 6, 1], 7, 720, 5040 [12, 9, 8, 7, 4, 3], 13, 720, 9360 [12, 9, 8, 6, 4, 4], 14, 360, 5040 [12, 9, 7, 7, 7, 1], 30, 120, 3600 [12, 9, 7, 7, 6, 2], 14, 360, 5040 [12, 9, 7, 7, 5, 3], 20, 360, 7200 [12, 9, 7, 7, 4, 4], 63, 180, 11340 [12, 9, 7, 6, 5, 4], 26, 720, 18720 [12, 8, 7, 7, 6, 3], 20, 360, 7200 [12, 8, 7, 6, 6, 4], 22, 360, 7920 [12, 7, 7, 7, 7, 3], 66, 30, 1980 [12, 7, 7, 7, 6, 4], 81, 120, 9720 [12, 7, 7, 6, 6, 5], 32, 180, 5760 [11, 10, 10, 9, 3, 0], 2, 360, 720 [11, 10, 10, 6, 6, 0], 4, 180, 720 [11, 10, 10, 6, 3, 3], 8, 180, 1440 [11, 10, 9, 9, 4, 0], 2, 360, 720 [11, 10, 9, 9, 3, 1], 2, 360, 720 [11, 10, 9, 7, 6, 0], 4, 720, 2880 [11, 10, 9, 7, 3, 3], 8, 360, 2880 [11, 10, 9, 6, 6, 1], 4, 360, 1440 [11, 10, 9, 6, 4, 3], 8, 720, 5760 [11, 10, 7, 6, 6, 3], 14, 360, 5040 [11, 10, 6, 6, 6, 4], 14, 120, 1680 [11, 9, 9, 9, 4, 1], 2, 120, 240 [11, 9, 9, 7, 7, 0], 4, 180, 720 [11, 9, 9, 7, 6, 1], 4, 360, 1440 [11, 9, 9, 7, 4, 3], 8, 360, 2880 [11, 9, 9, 6, 4, 4], 8, 180, 1440 [11, 9, 7, 7, 6, 3], 14, 360, 5040 [11, 9, 7, 6, 6, 4], 14, 360, 5040 [11, 7, 7, 6, 6, 6], 24, 60, 1440 [10, 10, 10, 10, 3, 0], 6, 30, 180 [10, 10, 10, 9, 4, 0], 9, 120, 1080 [10, 10, 10, 9, 3, 1], 9, 120, 1080 [10, 10, 10, 7, 6, 0], 15, 120, 1800 [10, 10, 10, 7, 3, 3], 30, 60, 1800 [10, 10, 10, 6, 6, 1], 18, 60, 1080 [10, 10, 10, 6, 4, 3], 33, 120, 3960 [10, 10, 9, 9, 5, 0], 4, 180, 720 [10, 10, 9, 9, 4, 1], 18, 180, 3240 [10, 10, 9, 9, 3, 2], 4, 180, 720 [10, 10, 9, 8, 6, 0], 6, 360, 2160 [10, 10, 9, 8, 3, 3], 12, 180, 2160 [10, 10, 9, 7, 7, 0], 19, 180, 3420 [10, 10, 9, 7, 6, 1], 24, 360, 8640 [10, 10, 9, 7, 4, 3], 43, 360, 15480 [10, 10, 9, 6, 6, 2], 8, 180, 1440 [10, 10, 9, 6, 5, 3], 14, 360, 5040 [10, 10, 9, 6, 4, 4], 52, 180, 9360 [10, 10, 8, 6, 6, 3], 20, 180, 3600 [10, 10, 7, 7, 6, 3], 63, 180, 11340 [10, 10, 7, 6, 6, 4], 72, 180, 12960 [10, 10, 6, 6, 6, 5], 24, 60, 1440 [10, 9, 9, 9, 5, 1], 12, 120, 1440 [10, 9, 9, 9, 4, 2], 14, 120, 1680 [10, 9, 9, 8, 7, 0], 8, 360, 2880 [10, 9, 9, 8, 6, 1], 10, 360, 3600 [10, 9, 9, 8, 4, 3], 18, 360, 6480 [10, 9, 9, 7, 7, 1], 34, 180, 6120 [10, 9, 9, 7, 6, 2], 14, 360, 5040 [10, 9, 9, 7, 5, 3], 22, 360, 7920 [10, 9, 9, 7, 4, 4], 72, 180, 12960 [10, 9, 9, 6, 5, 4], 30, 360, 10800 [10, 9, 8, 7, 6, 3], 26, 720, 18720 [10, 9, 8, 6, 6, 4], 30, 360, 10800 [10, 9, 7, 7, 7, 3], 81, 120, 9720 [10, 9, 7, 7, 6, 4], 96, 360, 34560 [10, 9, 7, 6, 6, 5], 36, 360, 12960 [10, 8, 7, 6, 6, 6], 42, 120, 5040 [10, 7, 7, 7, 6, 6], 132, 60, 7920 [9, 9, 9, 8, 7, 1], 12, 120, 1440 [9, 9, 9, 8, 4, 4], 24, 60, 1440 [9, 9, 9, 7, 7, 2], 24, 60, 1440 [9, 9, 9, 7, 5, 4], 42, 120, 5040 [9, 9, 8, 7, 7, 3], 32, 180, 5760 [9, 9, 8, 7, 6, 4], 36, 360, 12960 [9, 9, 7, 7, 7, 4], 132, 60, 7920 [9, 9, 7, 7, 6, 5], 52, 180, 9360 [9, 8, 7, 7, 6, 6], 52, 180, 9360 [9, 7, 7, 7, 7, 6], 174, 30, 5220 [15, 12, 9, 6, 1, 1], 1, 360, 360 [15, 12, 9, 4, 4, 0], 1, 360, 360 [15, 12, 9, 4, 3, 1], 1, 720, 720 [15, 12, 7, 7, 3, 0], 1, 360, 360 [15, 12, 7, 6, 4, 0], 1, 720, 720 [15, 12, 7, 6, 3, 1], 1, 720, 720 [15, 12, 7, 4, 3, 3], 2, 360, 720 [15, 12, 6, 6, 4, 1], 2, 360, 720 [15, 12, 6, 4, 4, 3], 3, 360, 1080 [15, 10, 10, 6, 3, 0], 1, 360, 360 [15, 10, 10, 3, 3, 3], 2, 60, 120 [15, 10, 9, 7, 3, 0], 1, 720, 720 [15, 10, 9, 6, 4, 0], 1, 720, 720 [15, 10, 9, 6, 3, 1], 1, 720, 720 [15, 10, 9, 4, 3, 3], 2, 360, 720 [15, 10, 7, 6, 6, 0], 2, 360, 720 [15, 10, 7, 6, 3, 3], 4, 360, 1440 [15, 10, 6, 6, 6, 1], 2, 120, 240 [15, 10, 6, 6, 4, 3], 4, 360, 1440 [15, 9, 9, 9, 1, 1], 2, 60, 120 [15, 9, 9, 7, 4, 0], 2, 360, 720 [15, 9, 9, 7, 3, 1], 2, 360, 720 [15, 9, 9, 6, 4, 1], 4, 360, 1440 [15, 9, 9, 4, 4, 3], 6, 180, 1080 [15, 9, 7, 7, 6, 0], 3, 360, 1080 [15, 9, 7, 7, 3, 3], 6, 180, 1080 [15, 9, 7, 6, 6, 1], 4, 360, 1440 [15, 9, 7, 6, 4, 3], 7, 720, 5040 [15, 9, 6, 6, 4, 4], 10, 180, 1800 [15, 7, 7, 6, 6, 3], 10, 180, 1800 [15, 7, 6, 6, 6, 4], 12, 120, 1440 [13, 13, 9, 6, 3, 0], 1, 360, 360 [13, 13, 9, 3, 3, 3], 2, 60, 120 [13, 13, 6, 6, 6, 0], 2, 60, 120 [13, 13, 6, 6, 3, 3], 4, 90, 360 [13, 12, 10, 6, 3, 0], 1, 720, 720 [13, 12, 10, 3, 3, 3], 2, 120, 240 [13, 12, 9, 7, 3, 0], 1, 720, 720 [13, 12, 9, 6, 4, 0], 1, 720, 720 [13, 12, 9, 6, 3, 1], 1, 720, 720 [13, 12, 9, 4, 3, 3], 2, 360, 720 [13, 12, 7, 6, 6, 0], 2, 360, 720 [13, 12, 7, 6, 3, 3], 4, 360, 1440 [13, 12, 6, 6, 6, 1], 2, 120, 240 [13, 12, 6, 6, 4, 3], 4, 360, 1440 [13, 10, 9, 9, 3, 0], 2, 360, 720 [13, 10, 9, 6, 6, 0], 4, 360, 1440 [13, 10, 9, 6, 3, 3], 8, 360, 2880 [13, 10, 6, 6, 6, 3], 14, 120, 1680 [13, 9, 9, 9, 4, 0], 2, 120, 240 [13, 9, 9, 9, 3, 1], 2, 120, 240 [13, 9, 9, 7, 6, 0], 4, 360, 1440 [13, 9, 9, 7, 3, 3], 8, 180, 1440 [13, 9, 9, 6, 6, 1], 4, 180, 720 [13, 9, 9, 6, 4, 3], 8, 360, 2880 [13, 9, 7, 6, 6, 3], 14, 360, 5040 [13, 9, 6, 6, 6, 4], 14, 120, 1680 [13, 7, 6, 6, 6, 6], 24, 30, 720 [12, 12, 12, 6, 1, 1], 2, 60, 120 [12, 12, 12, 4, 4, 0], 2, 60, 120 [12, 12, 12, 4, 3, 1], 2, 120, 240 [12, 12, 10, 7, 3, 0], 2, 360, 720 [12, 12, 10, 6, 4, 0], 2, 360, 720 [12, 12, 10, 6, 3, 1], 2, 360, 720 [12, 12, 10, 4, 3, 3], 4, 180, 720 [12, 12, 9, 9, 1, 1], 4, 90, 360 [12, 12, 9, 7, 4, 0], 4, 360, 1440 [12, 12, 9, 7, 3, 1], 4, 360, 1440 [12, 12, 9, 6, 4, 1], 8, 360, 2880 [12, 12, 9, 4, 4, 3], 12, 180, 2160 [12, 12, 7, 7, 6, 0], 6, 180, 1080 [12, 12, 7, 7, 3, 3], 12, 90, 1080 [12, 12, 7, 6, 6, 1], 8, 180, 1440 [12, 12, 7, 6, 4, 3], 14, 360, 5040 [12, 12, 6, 6, 4, 4], 20, 90, 1800 [12, 10, 10, 9, 3, 0], 3, 360, 1080 [12, 10, 10, 6, 6, 0], 6, 180, 1080 [12, 10, 10, 6, 3, 3], 12, 180, 2160 [12, 10, 9, 9, 4, 0], 4, 360, 1440 [12, 10, 9, 9, 3, 1], 4, 360, 1440 [12, 10, 9, 7, 6, 0], 7, 720, 5040 [12, 10, 9, 7, 3, 3], 14, 360, 5040 [12, 10, 9, 6, 6, 1], 8, 360, 2880 [12, 10, 9, 6, 4, 3], 15, 720, 10800 [12, 10, 7, 6, 6, 3], 24, 360, 8640 [12, 10, 6, 6, 6, 4], 26, 120, 3120 [12, 9, 9, 9, 4, 1], 14, 120, 1680 [12, 9, 9, 7, 7, 0], 10, 180, 1800 [12, 9, 9, 7, 6, 1], 14, 360, 5040 [12, 9, 9, 7, 4, 3], 24, 360, 8640 [12, 9, 9, 6, 4, 4], 34, 180, 6120 [12, 9, 7, 7, 6, 3], 33, 360, 11880 [12, 9, 7, 6, 6, 4], 40, 360, 14400 [12, 7, 7, 6, 6, 6], 54, 60, 3240 [10, 10, 9, 9, 6, 0], 10, 180, 1800 [10, 10, 9, 9, 3, 3], 20, 90, 1800 [10, 10, 9, 6, 6, 3], 34, 180, 6120 [10, 10, 6, 6, 6, 6], 56, 15, 840 [10, 9, 9, 9, 7, 0], 12, 120, 1440 [10, 9, 9, 9, 6, 1], 14, 120, 1680 [10, 9, 9, 9, 4, 3], 26, 120, 3120 [10, 9, 9, 7, 6, 3], 40, 360, 14400 [10, 9, 9, 6, 6, 4], 44, 180, 7920 [10, 9, 7, 6, 6, 6], 66, 120, 7920 [9, 9, 9, 9, 7, 1], 24, 30, 720 [9, 9, 9, 9, 4, 4], 56, 15, 840 [9, 9, 9, 7, 7, 3], 54, 60, 3240 [9, 9, 9, 7, 6, 4], 66, 120, 7920 [9, 9, 7, 7, 6, 6], 88, 90, 7920 [15, 12, 9, 6, 3, 0], 1, 720, 720 [15, 12, 9, 3, 3, 3], 2, 120, 240 [15, 12, 6, 6, 6, 0], 2, 120, 240 [15, 12, 6, 6, 3, 3], 4, 180, 720 [15, 9, 9, 9, 3, 0], 2, 120, 240 [15, 9, 9, 6, 6, 0], 4, 180, 720 [15, 9, 9, 6, 3, 3], 8, 180, 1440 [15, 9, 6, 6, 6, 3], 14, 120, 1680 [15, 6, 6, 6, 6, 6], 24, 6, 144 [12, 12, 12, 6, 3, 0], 2, 120, 240 [12, 12, 12, 3, 3, 3], 4, 20, 80 [12, 12, 9, 9, 3, 0], 4, 180, 720 [12, 12, 9, 6, 6, 0], 8, 180, 1440 [12, 12, 9, 6, 3, 3], 16, 180, 2880 [12, 12, 6, 6, 6, 3], 28, 60, 1680 [12, 9, 9, 9, 6, 0], 14, 120, 1680 [12, 9, 9, 9, 3, 3], 28, 60, 1680 [12, 9, 9, 6, 6, 3], 48, 180, 8640 [12, 9, 6, 6, 6, 6], 80, 30, 2400 [9, 9, 9, 9, 9, 0], 24, 6, 144 [9, 9, 9, 9, 6, 3], 80, 30, 2400 [9, 9, 9, 6, 6, 6], 132, 20, 2640 Sum of numbers in last column=, 14348907 The most probable vector is, [10, 8, 7, 6, 5, 4], which arises in, 88560, out of, 3, to the power, 15 that equals, 14348907, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 6, players The sole champion score-vector is, [10, 8, 7, 6, 5, 4] the corresponding record is, 88560 ----------------------------------------------------------------------------\ ----- 7, Players. Symmetric Function:, 7 / 7 \ --------' | --------' | ' | | |' | | 3 3 | | | | | | (a[i] + a[j] + 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, 6, 6, 6, 6, 6, 6], 1, 1, 1 [8, 6, 6, 6, 6, 6, 5], 1, 42, 42 [10, 6, 6, 6, 6, 5, 5], 1, 105, 105 [8, 8, 6, 6, 6, 6, 4], 1, 105, 105 [8, 8, 6, 6, 6, 5, 5], 2, 210, 420 [8, 7, 6, 6, 6, 6, 5], 1, 210, 210 [12, 6, 6, 6, 5, 5, 5], 1, 140, 140 [10, 8, 6, 6, 6, 5, 4], 1, 840, 840 [10, 8, 6, 6, 5, 5, 5], 3, 420, 1260 [10, 7, 6, 6, 6, 6, 4], 1, 210, 210 [10, 7, 6, 6, 6, 5, 5], 2, 420, 840 [9, 8, 6, 6, 6, 5, 5], 1, 420, 420 [8, 8, 8, 6, 6, 6, 3], 1, 140, 140 [8, 8, 8, 6, 6, 5, 4], 3, 420, 1260 [8, 8, 8, 6, 5, 5, 5], 6, 140, 840 [8, 8, 7, 6, 6, 6, 4], 2, 420, 840 [8, 8, 7, 6, 6, 5, 5], 4, 630, 2520 [8, 8, 6, 6, 6, 6, 5], 4, 105, 420 [8, 7, 7, 6, 6, 6, 5], 2, 420, 840 [7, 7, 7, 6, 6, 6, 6], 2, 35, 70 [14, 6, 6, 5, 5, 5, 5], 1, 105, 105 [12, 8, 6, 6, 5, 5, 4], 1, 1260, 1260 [12, 8, 6, 5, 5, 5, 5], 4, 210, 840 [12, 7, 6, 6, 6, 5, 4], 1, 840, 840 [12, 7, 6, 6, 5, 5, 5], 3, 420, 1260 [11, 8, 6, 6, 5, 5, 5], 1, 420, 420 [10, 10, 6, 6, 6, 4, 4], 1, 210, 210 [10, 10, 6, 6, 5, 5, 4], 2, 630, 1260 [10, 10, 6, 5, 5, 5, 5], 6, 105, 630 [10, 9, 6, 6, 6, 5, 4], 1, 840, 840 [10, 9, 6, 6, 5, 5, 5], 3, 420, 1260 [10, 8, 8, 6, 6, 5, 3], 1, 1260, 1260 [10, 8, 8, 6, 6, 4, 4], 2, 630, 1260 [10, 8, 8, 6, 5, 5, 4], 5, 1260, 6300 [10, 8, 8, 5, 5, 5, 5], 12, 105, 1260 [10, 8, 7, 6, 6, 6, 3], 1, 840, 840 [10, 8, 7, 6, 6, 5, 4], 4, 2520, 10080 [10, 8, 7, 6, 5, 5, 5], 9, 840, 7560 [10, 8, 6, 6, 6, 6, 4], 4, 210, 840 [10, 8, 6, 6, 6, 5, 5], 6, 420, 2520 [10, 7, 7, 6, 6, 6, 4], 3, 420, 1260 [10, 7, 7, 6, 6, 5, 5], 6, 630, 3780 [10, 7, 6, 6, 6, 6, 5], 4, 210, 840 [9, 8, 8, 6, 6, 5, 4], 2, 1260, 2520 [9, 8, 8, 6, 5, 5, 5], 6, 420, 2520 [9, 8, 7, 6, 6, 6, 4], 1, 840, 840 [9, 8, 7, 6, 6, 5, 5], 3, 1260, 3780 [9, 7, 7, 6, 6, 6, 5], 2, 420, 840 [8, 8, 8, 8, 6, 6, 2], 1, 105, 105 [8, 8, 8, 8, 6, 5, 3], 4, 210, 840 [8, 8, 8, 8, 6, 4, 4], 6, 105, 630 [8, 8, 8, 8, 5, 5, 4], 12, 105, 1260 [8, 8, 8, 7, 6, 6, 3], 3, 420, 1260 [8, 8, 8, 7, 6, 5, 4], 9, 840, 7560 [8, 8, 8, 7, 5, 5, 5], 18, 140, 2520 [8, 8, 8, 6, 6, 6, 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10, 10, 9, 8, 4, 3], 432, 2520, 1088640 [12, 10, 10, 9, 7, 7, 1], 164, 1260, 206640 [12, 10, 10, 9, 7, 6, 2], 472, 2520, 1189440 [12, 10, 10, 9, 7, 5, 3], 561, 2520, 1413720 [12, 10, 10, 9, 7, 4, 4], 364, 1260, 458640 [12, 10, 10, 9, 6, 6, 3], 645, 1260, 812700 [12, 10, 10, 9, 6, 5, 4], 821, 2520, 2068920 [12, 10, 10, 9, 5, 5, 5], 1346, 420, 565320 [12, 10, 10, 8, 8, 8, 0], 80, 420, 33600 [12, 10, 10, 8, 8, 7, 1], 376, 1260, 473760 [12, 10, 10, 8, 8, 6, 2], 852, 1260, 1073520 [12, 10, 10, 8, 8, 5, 3], 1092, 1260, 1375920 [12, 10, 10, 8, 8, 4, 4], 980, 630, 617400 [12, 10, 10, 8, 7, 7, 2], 634, 1260, 798840 [12, 10, 10, 8, 7, 6, 3], 749, 2520, 1887480 [12, 10, 10, 8, 7, 5, 4], 1192, 2520, 3003840 [12, 10, 10, 8, 6, 6, 4], 1085, 1260, 1367100 [12, 10, 10, 8, 6, 5, 5], 1880, 1260, 2368800 [12, 10, 10, 7, 7, 7, 3], 312, 420, 131040 [12, 10, 10, 7, 7, 6, 4], 528, 1260, 665280 [12, 10, 10, 7, 7, 5, 5], 1412, 630, 889560 [12, 10, 10, 7, 6, 6, 5], 1343, 1260, 1692180 [12, 10, 10, 6, 6, 6, 6], 1470, 105, 154350 [12, 10, 9, 9, 9, 7, 0], 67, 840, 56280 [12, 10, 9, 9, 9, 6, 1], 281, 840, 236040 [12, 10, 9, 9, 9, 5, 2], 474, 840, 398160 [12, 10, 9, 9, 9, 4, 3], 420, 840, 352800 [12, 10, 9, 9, 8, 8, 0], 102, 1260, 128520 [12, 10, 9, 9, 8, 7, 1], 376, 2520, 947520 [12, 10, 9, 9, 8, 6, 2], 682, 2520, 1718640 [12, 10, 9, 9, 8, 5, 3], 863, 2520, 2174760 [12, 10, 9, 9, 8, 4, 4], 857, 1260, 1079820 [12, 10, 9, 9, 7, 7, 2], 563, 1260, 709380 [12, 10, 9, 9, 7, 6, 3], 701, 2520, 1766520 [12, 10, 9, 9, 7, 5, 4], 1007, 2520, 2537640 [12, 10, 9, 9, 6, 6, 4], 971, 1260, 1223460 [12, 10, 9, 9, 6, 5, 5], 1432, 1260, 1804320 [12, 10, 9, 8, 8, 8, 1], 631, 840, 530040 [12, 10, 9, 8, 8, 7, 2], 1000, 2520, 2520000 [12, 10, 9, 8, 8, 6, 3], 1140, 2520, 2872800 [12, 10, 9, 8, 8, 5, 4], 1882, 2520, 4742640 [12, 10, 9, 8, 7, 7, 3], 843, 2520, 2124360 [12, 10, 9, 8, 7, 6, 4], 1281, 5040, 6456240 [12, 10, 9, 8, 7, 5, 5], 2210, 2520, 5569200 [12, 10, 9, 8, 6, 6, 5], 1930, 2520, 4863600 [12, 10, 9, 7, 7, 7, 4], 708, 840, 594720 [12, 10, 9, 7, 7, 6, 5], 1551, 2520, 3908520 [12, 10, 9, 7, 6, 6, 6], 1572, 840, 1320480 [12, 10, 8, 8, 8, 8, 2], 1674, 210, 351540 [12, 10, 8, 8, 8, 7, 3], 1709, 840, 1435560 [12, 10, 8, 8, 8, 6, 4], 2461, 840, 2067240 [12, 10, 8, 8, 8, 5, 5], 3769, 420, 1582980 [12, 10, 8, 8, 7, 7, 4], 2008, 1260, 2530080 [12, 10, 8, 8, 7, 6, 5], 2982, 2520, 7514640 [12, 10, 8, 8, 6, 6, 6], 2568, 420, 1078560 [12, 10, 8, 7, 7, 7, 5], 2396, 840, 2012640 [12, 10, 8, 7, 7, 6, 6], 2011, 1260, 2533860 [12, 10, 7, 7, 7, 7, 6], 1008, 210, 211680 [12, 9, 9, 9, 9, 8, 0], 120, 210, 25200 [12, 9, 9, 9, 9, 7, 1], 386, 210, 81060 [12, 9, 9, 9, 9, 6, 2], 564, 210, 118440 [12, 9, 9, 9, 9, 5, 3], 744, 210, 156240 [12, 9, 9, 9, 9, 4, 4], 897, 105, 94185 [12, 9, 9, 9, 8, 8, 1], 558, 420, 234360 [12, 9, 9, 9, 8, 7, 2], 798, 840, 670320 [12, 9, 9, 9, 8, 6, 3], 933, 840, 783720 [12, 9, 9, 9, 8, 5, 4], 1461, 840, 1227240 [12, 9, 9, 9, 7, 7, 3], 842, 420, 353640 [12, 9, 9, 9, 7, 6, 4], 1233, 840, 1035720 [12, 9, 9, 9, 7, 5, 5], 1680, 420, 705600 [12, 9, 9, 9, 6, 6, 5], 1572, 420, 660240 [12, 9, 9, 8, 8, 8, 2], 1206, 420, 506520 [12, 9, 9, 8, 8, 7, 3], 1292, 1260, 1627920 [12, 9, 9, 8, 8, 6, 4], 1840, 1260, 2318400 [12, 9, 9, 8, 8, 5, 5], 2554, 630, 1609020 [12, 9, 9, 8, 7, 7, 4], 1632, 1260, 2056320 [12, 9, 9, 8, 7, 6, 5], 2210, 2520, 5569200 [12, 9, 9, 8, 6, 6, 6], 1998, 420, 839160 [12, 9, 9, 7, 7, 7, 5], 1917, 420, 805140 [12, 9, 9, 7, 7, 6, 6], 1825, 630, 1149750 [12, 9, 8, 8, 8, 8, 3], 2064, 210, 433440 [12, 9, 8, 8, 8, 7, 4], 2985, 840, 2507400 [12, 9, 8, 8, 8, 6, 5], 3399, 840, 2855160 [12, 9, 8, 8, 7, 7, 5], 3558, 1260, 4483080 [12, 9, 8, 8, 7, 6, 6], 2832, 1260, 3568320 [12, 9, 8, 7, 7, 7, 6], 2379, 840, 1998360 [12, 9, 7, 7, 7, 7, 7], 1320, 42, 55440 [12, 8, 8, 8, 8, 8, 4], 4890, 42, 205380 [12, 8, 8, 8, 8, 7, 5], 5892, 210, 1237320 [12, 8, 8, 8, 8, 6, 6], 4464, 105, 468720 [12, 8, 8, 8, 7, 7, 6], 4598, 420, 1931160 [12, 8, 8, 7, 7, 7, 7], 3936, 105, 413280 [11, 11, 11, 11, 10, 1, 1], 6, 105, 630 [11, 11, 11, 11, 9, 2, 1], 72, 210, 15120 [11, 11, 11, 11, 8, 4, 0], 6, 210, 1260 [11, 11, 11, 11, 8, 3, 1], 174, 210, 36540 [11, 11, 11, 11, 8, 2, 2], 372, 105, 39060 [11, 11, 11, 11, 7, 5, 0], 48, 210, 10080 [11, 11, 11, 11, 7, 4, 1], 222, 210, 46620 [11, 11, 11, 11, 7, 3, 2], 400, 210, 84000 [11, 11, 11, 11, 6, 6, 0], 84, 105, 8820 [11, 11, 11, 11, 6, 5, 1], 462, 210, 97020 [11, 11, 11, 11, 6, 4, 2], 528, 210, 110880 [11, 11, 11, 11, 6, 3, 3], 252, 105, 26460 [11, 11, 11, 11, 5, 5, 2], 992, 105, 104160 [11, 11, 11, 11, 5, 4, 3], 622, 210, 130620 [11, 11, 11, 11, 4, 4, 4], 594, 35, 20790 [11, 11, 11, 10, 10, 3, 0], 2, 420, 840 [11, 11, 11, 10, 10, 2, 1], 42, 420, 17640 [11, 11, 11, 10, 9, 4, 0], 20, 840, 16800 [11, 11, 11, 10, 9, 3, 1], 137, 840, 115080 [11, 11, 11, 10, 9, 2, 2], 270, 420, 113400 [11, 11, 11, 10, 8, 5, 0], 57, 840, 47880 [11, 11, 11, 10, 8, 4, 1], 288, 840, 241920 [11, 11, 11, 10, 8, 3, 2], 561, 840, 471240 [11, 11, 11, 10, 7, 6, 0], 58, 840, 48720 [11, 11, 11, 10, 7, 5, 1], 408, 840, 342720 [11, 11, 11, 10, 7, 4, 2], 618, 840, 519120 [11, 11, 11, 10, 7, 3, 3], 512, 420, 215040 [11, 11, 11, 10, 6, 6, 1], 466, 420, 195720 [11, 11, 11, 10, 6, 5, 2], 1029, 840, 864360 [11, 11, 11, 10, 6, 4, 3], 662, 840, 556080 [11, 11, 11, 10, 5, 5, 3], 1282, 420, 538440 [11, 11, 11, 10, 5, 4, 4], 1134, 420, 476280 [11, 11, 11, 9, 9, 5, 0], 108, 420, 45360 [11, 11, 11, 9, 9, 4, 1], 332, 420, 139440 [11, 11, 11, 9, 9, 3, 2], 336, 420, 141120 [11, 11, 11, 9, 8, 6, 0], 135, 840, 113400 [11, 11, 11, 9, 8, 5, 1], 732, 840, 614880 [11, 11, 11, 9, 8, 4, 2], 870, 840, 730800 [11, 11, 11, 9, 8, 3, 3], 468, 420, 196560 [11, 11, 11, 9, 7, 7, 0], 78, 420, 32760 [11, 11, 11, 9, 7, 6, 1], 527, 840, 442680 [11, 11, 11, 9, 7, 5, 2], 1110, 840, 932400 [11, 11, 11, 9, 7, 4, 3], 761, 840, 639240 [11, 11, 11, 9, 6, 6, 2], 726, 420, 304920 [11, 11, 11, 9, 6, 5, 3], 948, 840, 796320 [11, 11, 11, 9, 6, 4, 4], 1172, 420, 492240 [11, 11, 11, 9, 5, 5, 4], 2022, 420, 849240 [11, 11, 11, 8, 8, 7, 0], 132, 420, 55440 [11, 11, 11, 8, 8, 6, 1], 993, 420, 417060 [11, 11, 11, 8, 8, 5, 2], 2046, 420, 859320 [11, 11, 11, 8, 8, 4, 3], 1323, 420, 555660 [11, 11, 11, 8, 7, 7, 1], 738, 420, 309960 [11, 11, 11, 8, 7, 6, 2], 1500, 840, 1260000 [11, 11, 11, 8, 7, 5, 3], 1848, 840, 1552320 [11, 11, 11, 8, 7, 4, 4], 1830, 420, 768600 [11, 11, 11, 8, 6, 6, 3], 1134, 420, 476280 [11, 11, 11, 8, 6, 5, 4], 2607, 840, 2189880 [11, 11, 11, 8, 5, 5, 5], 4464, 140, 624960 [11, 11, 11, 7, 7, 7, 2], 1350, 140, 189000 [11, 11, 11, 7, 7, 6, 3], 1346, 420, 565320 [11, 11, 11, 7, 7, 5, 4], 2418, 420, 1015560 [11, 11, 11, 7, 6, 6, 4], 1866, 420, 783720 [11, 11, 11, 7, 6, 5, 5], 3282, 420, 1378440 [11, 11, 11, 6, 6, 6, 5], 1998, 140, 279720 [11, 11, 10, 10, 10, 4, 0], 12, 420, 5040 [11, 11, 10, 10, 10, 3, 1], 90, 420, 37800 [11, 11, 10, 10, 10, 2, 2], 198, 210, 41580 [11, 11, 10, 10, 9, 5, 0], 54, 1260, 68040 [11, 11, 10, 10, 9, 4, 1], 226, 1260, 284760 [11, 11, 10, 10, 9, 3, 2], 430, 1260, 541800 [11, 11, 10, 10, 8, 6, 0], 70, 1260, 88200 [11, 11, 10, 10, 8, 5, 1], 462, 1260, 582120 [11, 11, 10, 10, 8, 4, 2], 767, 1260, 966420 [11, 11, 10, 10, 8, 3, 3], 746, 630, 469980 [11, 11, 10, 10, 7, 7, 0], 40, 630, 25200 [11, 11, 10, 10, 7, 6, 1], 402, 1260, 506520 [11, 11, 10, 10, 7, 5, 2], 994, 1260, 1252440 [11, 11, 10, 10, 7, 4, 3], 710, 1260, 894600 [11, 11, 10, 10, 6, 6, 2], 1074, 630, 676620 [11, 11, 10, 10, 6, 5, 3], 1336, 1260, 1683360 [11, 11, 10, 10, 6, 4, 4], 1068, 630, 672840 [11, 11, 10, 10, 5, 5, 4], 1908, 630, 1202040 [11, 11, 10, 9, 9, 6, 0], 102, 1260, 128520 [11, 11, 10, 9, 9, 5, 1], 526, 1260, 662760 [11, 11, 10, 9, 9, 4, 2], 660, 1260, 831600 [11, 11, 10, 9, 9, 3, 3], 452, 630, 284760 [11, 11, 10, 9, 8, 7, 0], 110, 2520, 277200 [11, 11, 10, 9, 8, 6, 1], 713, 2520, 1796760 [11, 11, 10, 9, 8, 5, 2], 1462, 2520, 3684240 [11, 11, 10, 9, 8, 4, 3], 1037, 2520, 2613240 [11, 11, 10, 9, 7, 7, 1], 540, 1260, 680400 [11, 11, 10, 9, 7, 6, 2], 1149, 2520, 2895480 [11, 11, 10, 9, 7, 5, 3], 1422, 2520, 3583440 [11, 11, 10, 9, 7, 4, 4], 1310, 1260, 1650600 [11, 11, 10, 9, 6, 6, 3], 1084, 1260, 1365840 [11, 11, 10, 9, 6, 5, 4], 1999, 2520, 5037480 [11, 11, 10, 9, 5, 5, 5], 3252, 420, 1365840 [11, 11, 10, 8, 8, 8, 0], 159, 420, 66780 [11, 11, 10, 8, 8, 7, 1], 900, 1260, 1134000 [11, 11, 10, 8, 8, 6, 2], 2058, 1260, 2593080 [11, 11, 10, 8, 8, 5, 3], 2553, 1260, 3216780 [11, 11, 10, 8, 8, 4, 4], 2346, 630, 1477980 [11, 11, 10, 8, 7, 7, 2], 1651, 1260, 2080260 [11, 11, 10, 8, 7, 6, 3], 1876, 2520, 4727520 [11, 11, 10, 8, 7, 5, 4], 3068, 2520, 7731360 [11, 11, 10, 8, 6, 6, 4], 2574, 1260, 3243240 [11, 11, 10, 8, 6, 5, 5], 4418, 1260, 5566680 [11, 11, 10, 7, 7, 7, 3], 1442, 420, 605640 [11, 11, 10, 7, 7, 6, 4], 2172, 1260, 2736720 [11, 11, 10, 7, 7, 5, 5], 3876, 630, 2441880 [11, 11, 10, 7, 6, 6, 5], 3324, 1260, 4188240 [11, 11, 10, 6, 6, 6, 6], 2400, 105, 252000 [11, 11, 9, 9, 9, 7, 0], 132, 420, 55440 [11, 11, 9, 9, 9, 6, 1], 516, 420, 216720 [11, 11, 9, 9, 9, 5, 2], 852, 420, 357840 [11, 11, 9, 9, 9, 4, 3], 744, 420, 312480 [11, 11, 9, 9, 8, 8, 0], 220, 630, 138600 [11, 11, 9, 9, 8, 7, 1], 862, 1260, 1086120 [11, 11, 9, 9, 8, 6, 2], 1136, 1260, 1431360 [11, 11, 9, 9, 8, 5, 3], 1478, 1260, 1862280 [11, 11, 9, 9, 8, 4, 4], 1914, 630, 1205820 [11, 11, 9, 9, 7, 7, 2], 1312, 630, 826560 [11, 11, 9, 9, 7, 6, 3], 1196, 1260, 1506960 [11, 11, 9, 9, 7, 5, 4], 2350, 1260, 2961000 [11, 11, 9, 9, 6, 6, 4], 1656, 630, 1043280 [11, 11, 9, 9, 6, 5, 5], 2438, 630, 1535940 [11, 11, 9, 8, 8, 8, 1], 1476, 420, 619920 [11, 11, 9, 8, 8, 7, 2], 2188, 1260, 2756880 [11, 11, 9, 8, 8, 6, 3], 1808, 1260, 2278080 [11, 11, 9, 8, 8, 5, 4], 3926, 1260, 4946760 [11, 11, 9, 8, 7, 7, 3], 2036, 1260, 2565360 [11, 11, 9, 8, 7, 6, 4], 2917, 2520, 7350840 [11, 11, 9, 8, 7, 5, 5], 4780, 1260, 6022800 [11, 11, 9, 8, 6, 6, 5], 3084, 1260, 3885840 [11, 11, 9, 7, 7, 7, 4], 2622, 420, 1101240 [11, 11, 9, 7, 7, 6, 5], 3680, 1260, 4636800 [11, 11, 9, 7, 6, 6, 6], 2604, 420, 1093680 [11, 11, 8, 8, 8, 8, 2], 3972, 105, 417060 [11, 11, 8, 8, 8, 7, 3], 3607, 420, 1514940 [11, 11, 8, 8, 8, 6, 4], 5004, 420, 2101680 [11, 11, 8, 8, 8, 5, 5], 8432, 210, 1770720 [11, 11, 8, 8, 7, 7, 4], 4790, 630, 3017700 [11, 11, 8, 8, 7, 6, 5], 6255, 1260, 7881300 [11, 11, 8, 8, 6, 6, 6], 3786, 210, 795060 [11, 11, 8, 7, 7, 7, 5], 6016, 420, 2526720 [11, 11, 8, 7, 7, 6, 6], 4666, 630, 2939580 [11, 11, 7, 7, 7, 7, 6], 4184, 105, 439320 [11, 10, 10, 10, 10, 5, 0], 24, 210, 5040 [11, 10, 10, 10, 10, 4, 1], 120, 210, 25200 [11, 10, 10, 10, 10, 3, 2], 308, 210, 64680 [11, 10, 10, 10, 9, 6, 0], 42, 840, 35280 [11, 10, 10, 10, 9, 5, 1], 324, 840, 272160 [11, 10, 10, 10, 9, 4, 2], 525, 840, 441000 [11, 10, 10, 10, 9, 3, 3], 495, 420, 207900 [11, 10, 10, 10, 8, 7, 0], 48, 840, 40320 [11, 10, 10, 10, 8, 6, 1], 453, 840, 380520 [11, 10, 10, 10, 8, 5, 2], 1129, 840, 948360 [11, 10, 10, 10, 8, 4, 3], 893, 840, 750120 [11, 10, 10, 10, 7, 7, 1], 300, 420, 126000 [11, 10, 10, 10, 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[15, 12, 10, 9, 9, 7, 0], 40, 2520, 100800 [15, 12, 10, 9, 9, 6, 1], 48, 2520, 120960 [15, 12, 10, 9, 9, 4, 3], 88, 2520, 221760 [15, 12, 10, 9, 7, 6, 3], 133, 5040, 670320 [15, 12, 10, 9, 6, 6, 4], 148, 2520, 372960 [15, 12, 10, 7, 6, 6, 6], 218, 840, 183120 [15, 12, 9, 9, 9, 7, 1], 80, 840, 67200 [15, 12, 9, 9, 9, 4, 4], 186, 420, 78120 [15, 12, 9, 9, 7, 7, 3], 178, 1260, 224280 [15, 12, 9, 9, 7, 6, 4], 218, 2520, 549360 [15, 12, 9, 7, 7, 6, 6], 288, 1260, 362880 [15, 10, 10, 9, 9, 9, 0], 54, 420, 22680 [15, 10, 10, 9, 9, 6, 3], 178, 1260, 224280 [15, 10, 10, 9, 6, 6, 6], 290, 420, 121800 [15, 10, 9, 9, 9, 9, 1], 80, 210, 16800 [15, 10, 9, 9, 9, 7, 3], 212, 840, 178080 [15, 10, 9, 9, 9, 6, 4], 238, 840, 199920 [15, 10, 9, 9, 7, 6, 6], 344, 1260, 433440 [15, 9, 9, 9, 9, 7, 4], 344, 210, 72240 [15, 9, 9, 9, 7, 7, 6], 450, 420, 189000 [13, 13, 12, 12, 9, 3, 0], 10, 1260, 12600 [13, 13, 12, 12, 6, 6, 0], 20, 630, 12600 [13, 13, 12, 12, 6, 3, 3], 40, 630, 25200 [13, 13, 12, 9, 9, 6, 0], 34, 1260, 42840 [13, 13, 12, 9, 9, 3, 3], 68, 630, 42840 [13, 13, 12, 9, 6, 6, 3], 116, 1260, 146160 [13, 13, 12, 6, 6, 6, 6], 192, 105, 20160 [13, 13, 9, 9, 9, 9, 0], 56, 105, 5880 [13, 13, 9, 9, 9, 6, 3], 186, 420, 78120 [13, 13, 9, 9, 6, 6, 6], 304, 210, 63840 [13, 12, 12, 12, 10, 3, 0], 12, 840, 10080 [13, 12, 12, 12, 9, 4, 0], 14, 840, 11760 [13, 12, 12, 12, 9, 3, 1], 14, 840, 11760 [13, 12, 12, 12, 7, 6, 0], 26, 840, 21840 [13, 12, 12, 12, 7, 3, 3], 52, 420, 21840 [13, 12, 12, 12, 6, 6, 1], 28, 420, 11760 [13, 12, 12, 12, 6, 4, 3], 54, 840, 45360 [13, 12, 12, 10, 9, 6, 0], 40, 2520, 100800 [13, 12, 12, 10, 9, 3, 3], 80, 1260, 100800 [13, 12, 12, 10, 6, 6, 3], 136, 1260, 171360 [13, 12, 12, 9, 9, 7, 0], 44, 1260, 55440 [13, 12, 12, 9, 9, 6, 1], 48, 1260, 60480 [13, 12, 12, 9, 9, 4, 3], 92, 1260, 115920 [13, 12, 12, 9, 7, 6, 3], 148, 2520, 372960 [13, 12, 12, 9, 6, 6, 4], 156, 1260, 196560 [13, 12, 12, 7, 6, 6, 6], 244, 420, 102480 [13, 12, 10, 9, 9, 9, 0], 66, 840, 55440 [13, 12, 10, 9, 9, 6, 3], 218, 2520, 549360 [13, 12, 10, 9, 6, 6, 6], 356, 840, 299040 [13, 12, 9, 9, 9, 9, 1], 80, 210, 16800 [13, 12, 9, 9, 9, 7, 3], 238, 840, 199920 [13, 12, 9, 9, 9, 6, 4], 252, 840, 211680 [13, 12, 9, 9, 7, 6, 6], 388, 1260, 488880 [13, 10, 9, 9, 9, 9, 3], 344, 210, 72240 [13, 10, 9, 9, 9, 6, 6], 556, 420, 233520 [13, 9, 9, 9, 9, 9, 4], 400, 42, 16800 [13, 9, 9, 9, 9, 7, 6], 608, 210, 127680 [12, 12, 12, 12, 12, 1, 1], 24, 21, 504 [12, 12, 12, 12, 10, 4, 0], 24, 210, 5040 [12, 12, 12, 12, 10, 3, 1], 24, 210, 5040 [12, 12, 12, 12, 9, 4, 1], 80, 210, 16800 [12, 12, 12, 12, 7, 7, 0], 56, 105, 5880 [12, 12, 12, 12, 7, 6, 1], 80, 210, 16800 [12, 12, 12, 12, 7, 4, 3], 136, 210, 28560 [12, 12, 12, 12, 6, 4, 4], 192, 105, 20160 [12, 12, 12, 10, 10, 6, 0], 54, 420, 22680 [12, 12, 12, 10, 10, 3, 3], 108, 210, 22680 [12, 12, 12, 10, 9, 7, 0], 66, 840, 55440 [12, 12, 12, 10, 9, 6, 1], 80, 840, 67200 [12, 12, 12, 10, 9, 4, 3], 146, 840, 122640 [12, 12, 12, 10, 7, 6, 3], 218, 840, 183120 [12, 12, 12, 10, 6, 6, 4], 244, 420, 102480 [12, 12, 12, 9, 9, 7, 1], 132, 420, 55440 [12, 12, 12, 9, 9, 4, 4], 304, 210, 63840 [12, 12, 12, 9, 7, 7, 3], 290, 420, 121800 [12, 12, 12, 9, 7, 6, 4], 356, 840, 299040 [12, 12, 12, 7, 7, 6, 6], 468, 210, 98280 [12, 12, 10, 10, 9, 9, 0], 88, 630, 55440 [12, 12, 10, 10, 9, 6, 3], 288, 1260, 362880 [12, 12, 10, 10, 6, 6, 6], 468, 210, 98280 [12, 12, 10, 9, 9, 9, 1], 132, 420, 55440 [12, 12, 10, 9, 9, 7, 3], 344, 1260, 433440 [12, 12, 10, 9, 9, 6, 4], 388, 1260, 488880 [12, 12, 10, 9, 7, 6, 6], 556, 1260, 700560 [12, 12, 9, 9, 9, 7, 4], 556, 420, 233520 [12, 12, 9, 9, 7, 7, 6], 724, 630, 456120 [12, 10, 10, 9, 9, 9, 3], 450, 420, 189000 [12, 10, 10, 9, 9, 6, 6], 724, 630, 456120 [12, 10, 9, 9, 9, 9, 4], 608, 210, 127680 [12, 10, 9, 9, 9, 7, 6], 860, 840, 722400 [12, 9, 9, 9, 9, 7, 7], 1112, 105, 116760 [10, 10, 9, 9, 9, 9, 6], 1112, 105, 116760 [10, 9, 9, 9, 9, 9, 7], 1320, 42, 55440 [18, 15, 12, 9, 6, 3, 0], 1, 5040, 5040 [18, 15, 12, 9, 3, 3, 3], 2, 840, 1680 [18, 15, 12, 6, 6, 6, 0], 2, 840, 1680 [18, 15, 12, 6, 6, 3, 3], 4, 1260, 5040 [18, 15, 9, 9, 9, 3, 0], 2, 840, 1680 [18, 15, 9, 9, 6, 6, 0], 4, 1260, 5040 [18, 15, 9, 9, 6, 3, 3], 8, 1260, 10080 [18, 15, 9, 6, 6, 6, 3], 14, 840, 11760 [18, 15, 6, 6, 6, 6, 6], 24, 42, 1008 [18, 12, 12, 12, 6, 3, 0], 2, 840, 1680 [18, 12, 12, 12, 3, 3, 3], 4, 140, 560 [18, 12, 12, 9, 9, 3, 0], 4, 1260, 5040 [18, 12, 12, 9, 6, 6, 0], 8, 1260, 10080 [18, 12, 12, 9, 6, 3, 3], 16, 1260, 20160 [18, 12, 12, 6, 6, 6, 3], 28, 420, 11760 [18, 12, 9, 9, 9, 6, 0], 14, 840, 11760 [18, 12, 9, 9, 9, 3, 3], 28, 420, 11760 [18, 12, 9, 9, 6, 6, 3], 48, 1260, 60480 [18, 12, 9, 6, 6, 6, 6], 80, 210, 16800 [18, 9, 9, 9, 9, 9, 0], 24, 42, 1008 [18, 9, 9, 9, 9, 6, 3], 80, 210, 16800 [18, 9, 9, 9, 6, 6, 6], 132, 140, 18480 [15, 15, 15, 9, 6, 3, 0], 2, 840, 1680 [15, 15, 15, 9, 3, 3, 3], 4, 140, 560 [15, 15, 15, 6, 6, 6, 0], 4, 140, 560 [15, 15, 15, 6, 6, 3, 3], 8, 210, 1680 [15, 15, 12, 12, 6, 3, 0], 4, 1260, 5040 [15, 15, 12, 12, 3, 3, 3], 8, 210, 1680 [15, 15, 12, 9, 9, 3, 0], 8, 1260, 10080 [15, 15, 12, 9, 6, 6, 0], 16, 1260, 20160 [15, 15, 12, 9, 6, 3, 3], 32, 1260, 40320 [15, 15, 12, 6, 6, 6, 3], 56, 420, 23520 [15, 15, 9, 9, 9, 6, 0], 28, 420, 11760 [15, 15, 9, 9, 9, 3, 3], 56, 210, 11760 [15, 15, 9, 9, 6, 6, 3], 96, 630, 60480 [15, 15, 9, 6, 6, 6, 6], 160, 105, 16800 [15, 12, 12, 12, 9, 3, 0], 14, 840, 11760 [15, 12, 12, 12, 6, 6, 0], 28, 420, 11760 [15, 12, 12, 12, 6, 3, 3], 56, 420, 23520 [15, 12, 12, 9, 9, 6, 0], 48, 1260, 60480 [15, 12, 12, 9, 9, 3, 3], 96, 630, 60480 [15, 12, 12, 9, 6, 6, 3], 164, 1260, 206640 [15, 12, 12, 6, 6, 6, 6], 272, 105, 28560 [15, 12, 9, 9, 9, 9, 0], 80, 210, 16800 [15, 12, 9, 9, 9, 6, 3], 266, 840, 223440 [15, 12, 9, 9, 6, 6, 6], 436, 420, 183120 [15, 9, 9, 9, 9, 9, 3], 424, 42, 17808 [15, 9, 9, 9, 9, 6, 6], 688, 105, 72240 [12, 12, 12, 12, 12, 3, 0], 24, 42, 1008 [12, 12, 12, 12, 9, 6, 0], 80, 210, 16800 [12, 12, 12, 12, 9, 3, 3], 160, 105, 16800 [12, 12, 12, 12, 6, 6, 3], 272, 105, 28560 [12, 12, 12, 9, 9, 9, 0], 132, 140, 18480 [12, 12, 12, 9, 9, 6, 3], 436, 420, 183120 [12, 12, 12, 9, 6, 6, 6], 712, 140, 99680 [12, 12, 9, 9, 9, 9, 3], 688, 105, 72240 [12, 12, 9, 9, 9, 6, 6], 1112, 210, 233520 [12, 9, 9, 9, 9, 9, 6], 1720, 42, 72240 [9, 9, 9, 9, 9, 9, 9], 2640, 1, 2640 Sum of numbers in last column=, 10460353203 The most probable vector is, [11, 10, 9, 8, 7, 6, 5], which arises in, 23758560, out of, 3, to the power, 21 that equals, 10460353203, events. ----------------------------------------------------------------------------\ ----- ------------------------------------------ In a torunament with, 7, players The sole champion score-vector is, [11, 10, 9, 8, 7, 6, 5] the corresponding record is, 23758560 ------------------------------------------ In conclusion, the n=2 thorough n=, 7, of the champions are [6, 36, 960, 88560, 23758560] ------------------------------------------ This ends this article, that took, 146.093, seconds to produce.