All the Polynomial Expressions for the Number of Round-Robin Tournaments whose excess is <=, 6 By Shalosh B. Ekhad ---------------------------------------------------- Chapter Number, 0 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 0 If the Goals For Vector Was [r, r, r, r] ----------------------------------------------------------------- Proposition No. , 0.1, : The number of possible Soccer Tournaments with Goals For vector, [r, r, r, r] and Goals Against Vector, [r, r, r, r], is given by 2 (2 r + 3) (r + 2) (r + 1) (r + 3 r + 5) ---------------------------------------- 30 and in Maple input form 1/30*(2*r+3)*(r+2)*(r+1)*(r^2+3*r+5) The first few terms, starting at, 1, are [9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, 189202, 242802, 307923] This concludes this chapter, that took, 0.104, seconds to produce. ---------------------------------------------------- Chapter Number, 1 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 1 If the Goals For Vector Was [r + 1, r, r, r] ----------------------------------------------------------------- Proposition No. , 1.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r, r, r] and Goals Against Vector, [r + 1, r, r, r], is given by r (4 r + 11) (r + 3) (r + 2) (r + 1) ------------------------------------ 60 and in Maple input form 1/60*r*(4*r+11)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [6, 38, 138, 378, 868, 1764, 3276, 5676, 9306, 14586, 22022, 32214, 45864, 63784, 86904, 116280, 153102, 198702, 254562, 322322] This concludes this chapter, that took, 0.111, seconds to produce. ---------------------------------------------------- Chapter Number, 2 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 2 If the Goals For Vector Was [r + 1, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 2.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r, r] and Goals Against Vector, [r + 1, r + 1, r, r], is given by 3 2 (r + 2) (r + 1) (2 r + 14 r + 24 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+14*r^2+24*r+15) The first few terms, starting at, 1, are [11, 54, 178, 463, 1029, 2044, 3732, 6381, 10351, 16082, 24102, 35035, 49609, 68664, 93160, 124185, 162963, 210862, 269402, 340263] ----------------------------------------------------------------- Proposition No. , 2.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r, r] and Goals Against Vector, [r + 2, r, r, r], is given by 2 r (r + 2) (r + 1) (r + 7 r + 7) -------------------------------- 15 and in Maple input form 1/15*r*(r+2)*(r+1)*(r^2+7*r+7) The first few terms, starting at, 1, are [6, 40, 148, 408, 938, 1904, 3528, 6096, 9966, 15576, 23452, 34216, 48594, 67424, 91664, 122400, 160854, 208392, 266532, 336952] If the Goals For Vector Was [r + 2, r, r, r] ----------------------------------------------------------------- Proposition No. , 2.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r, r, r] and Goals Against Vector, [r + 1, r + 1, r, r], is given by 2 r (r + 2) (r + 1) (r + 7 r + 7) -------------------------------- 15 and in Maple input form 1/15*r*(r+2)*(r+1)*(r^2+7*r+7) The first few terms, starting at, 1, are [6, 40, 148, 408, 938, 1904, 3528, 6096, 9966, 15576, 23452, 34216, 48594, 67424, 91664, 122400, 160854, 208392, 266532, 336952] ----------------------------------------------------------------- Proposition No. , 2.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r, r, r] and Goals Against Vector, [r + 2, r, r, r], is given by 3 2 r (r + 1) (2 r + 18 r + 22 r - 27) ------------------------------------ 30 and in Maple input form 1/30*r*(r+1)*(2*r^3+18*r^2+22*r-27) The first few terms, starting at, 1, are [1, 21, 102, 318, 783, 1659, 3164, 5580, 9261, 14641, 22242, 32682, 46683, 65079, 88824, 119000, 156825, 203661, 261022, 330582] This concludes this chapter, that took, 0.270, seconds to produce. ---------------------------------------------------- Chapter Number, 3 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 3 If the Goals For Vector Was [r + 1, r + 1, r + 1, r] ----------------------------------------------------------------- Proposition No. , 3.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r], is given by (4 r + 5) (r + 4) (r + 3) (r + 2) (r + 1) ----------------------------------------- 60 and in Maple input form 1/60*(4*r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [18, 78, 238, 588, 1260, 2436, 4356, 7326, 11726, 18018, 26754, 38584, 54264, 74664, 100776, 133722, 174762, 225302, 286902, 361284] ----------------------------------------------------------------- Proposition No. , 3.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 1, r, r], is given by 3 2 (r + 2) (r + 1) (4 r + 33 r + 63 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+33*r^2+63*r+30) The first few terms, starting at, 1, are [13, 64, 208, 533, 1169, 2296, 4152, 7041, 11341, 17512, 26104, 37765, 53249, 73424, 99280, 131937, 172653, 222832, 284032, 357973] ----------------------------------------------------------------- Proposition No. , 3.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r, r, r], is given by 2 r (r + 2) (r + 1) (4 r + 33 r + 23) ------------------------------------ 60 and in Maple input form 1/60*r*(r+2)*(r+1)*(4*r^2+33*r+23) The first few terms, starting at, 1, are [6, 42, 158, 438, 1008, 2044, 3780, 6516, 10626, 16566, 24882, 36218, 51324, 71064, 96424, 128520, 168606, 218082, 278502, 351582] If the Goals For Vector Was [r + 2, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 3.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r], is given by 3 2 (r + 2) (r + 1) (4 r + 33 r + 63 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+33*r^2+63*r+30) The first few terms, starting at, 1, are [13, 64, 208, 533, 1169, 2296, 4152, 7041, 11341, 17512, 26104, 37765, 53249, 73424, 99280, 131937, 172653, 222832, 284032, 357973] ----------------------------------------------------------------- Proposition No. , 3.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r] and Goals Against Vector, [r + 2, r + 1, r, r], is given by 2 r (r + 2) (r + 1) (4 r + 33 r + 23) ------------------------------------ 60 and in Maple input form 1/60*r*(r+2)*(r+1)*(4*r^2+33*r+23) The first few terms, starting at, 1, are [6, 42, 158, 438, 1008, 2044, 3780, 6516, 10626, 16566, 24882, 36218, 51324, 71064, 96424, 128520, 168606, 218082, 278502, 351582] ----------------------------------------------------------------- Proposition No. , 3.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r] and Goals Against Vector, [r + 3, r, r, r], is given by 4 3 2 r (4 r + 45 r + 80 r - 45 r - 24) ------------------------------------ 60 and in Maple input form 1/60*r*(4*r^4+45*r^3+80*r^2-45*r-24) The first few terms, starting at, 1, are [1, 21, 105, 332, 823, 1749, 3339, 5888, 9765, 15421, 23397, 34332, 48971, 68173, 92919, 124320, 163625, 212229, 271681, 343692] If the Goals For Vector Was [r + 3, r, r, r] ----------------------------------------------------------------- Proposition No. , 3.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r], is given by 2 r (r + 2) (r + 1) (4 r + 33 r + 23) ------------------------------------ 60 and in Maple input form 1/60*r*(r+2)*(r+1)*(4*r^2+33*r+23) The first few terms, starting at, 1, are [6, 42, 158, 438, 1008, 2044, 3780, 6516, 10626, 16566, 24882, 36218, 51324, 71064, 96424, 128520, 168606, 218082, 278502, 351582] ----------------------------------------------------------------- Proposition No. , 3.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r] and Goals Against Vector, [r + 2, r + 1, r, r], is given by 4 3 2 r (4 r + 45 r + 80 r - 45 r - 24) ------------------------------------ 60 and in Maple input form 1/60*r*(4*r^4+45*r^3+80*r^2-45*r-24) The first few terms, starting at, 1, are [1, 21, 105, 332, 823, 1749, 3339, 5888, 9765, 15421, 23397, 34332, 48971, 68173, 92919, 124320, 163625, 212229, 271681, 343692] ----------------------------------------------------------------- Proposition No. , 3.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r] and Goals Against Vector, [r + 3, r, r, r], is given by 3 2 r (r - 1) (4 r + 49 r + 59 r - 166) ------------------------------------- 60 and in Maple input form 1/60*r*(r-1)*(4*r^3+49*r^2+59*r-166) The first few terms, starting at, 1, are [0, 6, 56, 222, 618, 1408, 2814, 5124, 8700, 13986, 21516, 31922, 45942, 64428, 88354, 118824, 157080, 204510, 262656, 333222] This concludes this chapter, that took, 0.510, seconds to produce. ---------------------------------------------------- Chapter Number, 4 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 4 If the Goals For Vector Was [r + 1, r + 1, r + 1, r + 1] the smallest entry of, [1, 1, 1, 1], [1, 1, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 4.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 2, r + 1, r + 1, r], is given by (2 r + 5) (r + 4) (r + 3) (r + 2) (r + 1) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [28, 108, 308, 728, 1512, 2856, 5016, 8316, 13156, 20020, 29484, 42224, 59024, 80784, 108528, 143412, 186732, 239932, 304612, 382536] ----------------------------------------------------------------- Proposition No. , 4.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 2, r + 2, r, r], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 13 r + 10) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+13*r+10) The first few terms, starting at, 1, are [20, 88, 268, 658, 1400, 2688, 4776, 7986, 12716, 19448, 28756, 41314, 57904, 79424, 106896, 141474, 184452, 237272, 301532, 378994] ----------------------------------------------------------------- Proposition No. , 4.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 1, r, r], is given by 2 (2 r + 1) (r + 2) (r + 1) (r + 9 r + 15) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+1)*(r+2)*(r+1)*(r^2+9*r+15) The first few terms, starting at, 1, are [15, 74, 238, 603, 1309, 2548, 4572, 7701, 12331, 18942, 28106, 40495, 56889, 78184, 105400, 139689, 182343, 234802, 298662, 375683] ----------------------------------------------------------------- Proposition No. , 4.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 4, r, r, r], is given by r (r + 9) (2 r + 1) (r + 2) (r + 1) ----------------------------------- 30 and in Maple input form 1/30*r*(r+9)*(2*r+1)*(r+2)*(r+1) The first few terms, starting at, 1, are [6, 44, 168, 468, 1078, 2184, 4032, 6936, 11286, 17556, 26312, 38220, 54054, 74704, 101184, 134640, 176358, 227772, 290472, 366212] If the Goals For Vector Was [r + 2, r + 1, r + 1, r] ----------------------------------------------------------------- Proposition No. , 4.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 1], is given by (2 r + 5) (r + 4) (r + 3) (r + 2) (r + 1) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [28, 108, 308, 728, 1512, 2856, 5016, 8316, 13156, 20020, 29484, 42224, 59024, 80784, 108528, 143412, 186732, 239932, 304612, 382536] ----------------------------------------------------------------- Proposition No. , 4.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r], is given by 2 (2 r + 1) (r + 2) (r + 1) (r + 9 r + 15) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+1)*(r+2)*(r+1)*(r^2+9*r+15) The first few terms, starting at, 1, are [15, 74, 238, 603, 1309, 2548, 4572, 7701, 12331, 18942, 28106, 40495, 56889, 78184, 105400, 139689, 182343, 234802, 298662, 375683] ----------------------------------------------------------------- Proposition No. , 4.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r, r], is given by 3 2 (r + 2) (r + 1) (2 r + 19 r + 29 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+19*r^2+29*r+15) The first few terms, starting at, 1, are [13, 66, 218, 563, 1239, 2436, 4404, 7461, 12001, 18502, 27534, 39767, 55979, 77064, 104040, 138057, 180405, 232522, 296002, 372603] ----------------------------------------------------------------- Proposition No. , 4.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 1, r, r], is given by 3 2 r (r + 1) (2 r + 23 r + 52 r + 13) ------------------------------------ 30 and in Maple input form 1/30*r*(r+1)*(2*r^3+23*r^2+52*r+13) The first few terms, starting at, 1, are [6, 45, 172, 478, 1098, 2219, 4088, 7020, 11406, 17721, 26532, 38506, 54418, 75159, 101744, 135320, 177174, 228741, 291612, 367542] ----------------------------------------------------------------- Proposition No. , 4.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r + 4, r, r, r], is given by 4 3 2 r (2 r + 25 r + 40 r - 40 r + 3) ----------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+25*r^3+40*r^2-40*r+3) The first few terms, starting at, 1, are [1, 21, 108, 346, 863, 1839, 3514, 6196, 10269, 16201, 24552, 35982, 51259, 71267, 97014, 129640, 170425, 220797, 282340, 356802] If the Goals For Vector Was [r + 2, r + 2, r, r] ----------------------------------------------------------------- Proposition No. , 4.10, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 13 r + 10) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+13*r+10) The first few terms, starting at, 1, are [20, 88, 268, 658, 1400, 2688, 4776, 7986, 12716, 19448, 28756, 41314, 57904, 79424, 106896, 141474, 184452, 237272, 301532, 378994] ----------------------------------------------------------------- Proposition No. , 4.11, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r], is given by 3 2 (r + 2) (r + 1) (2 r + 19 r + 29 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+19*r^2+29*r+15) The first few terms, starting at, 1, are [13, 66, 218, 563, 1239, 2436, 4404, 7461, 12001, 18502, 27534, 39767, 55979, 77064, 104040, 138057, 180405, 232522, 296002, 372603] ----------------------------------------------------------------- Proposition No. , 4.12, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r + 2, r + 2, r, r], is given by 4 3 2 (r + 1) (2 r + 23 r + 47 r + 63 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+1)*(2*r^4+23*r^3+47*r^2+63*r+30) The first few terms, starting at, 1, are [11, 56, 190, 503, 1129, 2254, 4124, 7053, 11431, 17732, 26522, 38467, 54341, 75034, 101560, 135065, 176835, 228304, 291062, 366863] ----------------------------------------------------------------- Proposition No. , 4.13, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r + 3, r + 1, r, r], is given by 3 2 r (r + 1) (2 r + 23 r + 37 r + 28) ------------------------------------ 30 and in Maple input form 1/30*r*(r+1)*(2*r^3+23*r^2+37*r+28) The first few terms, starting at, 1, are [6, 42, 160, 448, 1038, 2114, 3920, 6768, 11046, 17226, 25872, 37648, 53326, 73794, 100064, 133280, 174726, 225834, 288192, 363552] ----------------------------------------------------------------- Proposition No. , 4.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r + 4, r, r, r], is given by 5 4 3 2 14 1/15 r + 5/6 r + r - 5/6 r + -- r - 1 15 and in Maple input form 1/15*r^5+5/6*r^4+r^3-5/6*r^2+14/15*r-1 The first few terms, starting at, 1, are [1, 21, 105, 335, 837, 1789, 3429, 6063, 10073, 15925, 24177, 35487, 50621, 70461, 96013, 128415, 168945, 219029, 280249, 354351] If the Goals For Vector Was [r + 3, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 4.15, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 1], is given by 2 (2 r + 1) (r + 2) (r + 1) (r + 9 r + 15) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+1)*(r+2)*(r+1)*(r^2+9*r+15) The first few terms, starting at, 1, are [15, 74, 238, 603, 1309, 2548, 4572, 7701, 12331, 18942, 28106, 40495, 56889, 78184, 105400, 139689, 182343, 234802, 298662, 375683] ----------------------------------------------------------------- Proposition No. , 4.16, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r], is given by 3 2 r (r + 1) (2 r + 23 r + 52 r + 13) ------------------------------------ 30 and in Maple input form 1/30*r*(r+1)*(2*r^3+23*r^2+52*r+13) The first few terms, starting at, 1, are [6, 45, 172, 478, 1098, 2219, 4088, 7020, 11406, 17721, 26532, 38506, 54418, 75159, 101744, 135320, 177174, 228741, 291612, 367542] ----------------------------------------------------------------- Proposition No. , 4.17, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r] and Goals Against Vector, [r + 2, r + 2, r, r], is given by 3 2 r (r + 1) (2 r + 23 r + 37 r + 28) ------------------------------------ 30 and in Maple input form 1/30*r*(r+1)*(2*r^3+23*r^2+37*r+28) The first few terms, starting at, 1, are [6, 42, 160, 448, 1038, 2114, 3920, 6768, 11046, 17226, 25872, 37648, 53326, 73794, 100064, 133280, 174726, 225834, 288192, 363552] ----------------------------------------------------------------- Proposition No. , 4.18, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r] and Goals Against Vector, [r + 3, r + 1, r, r], is given by 4 3 2 r (2 r + 25 r + 40 r - 40 r + 3) ----------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+25*r^3+40*r^2-40*r+3) The first few terms, starting at, 1, are [1, 21, 108, 346, 863, 1839, 3514, 6196, 10269, 16201, 24552, 35982, 51259, 71267, 97014, 129640, 170425, 220797, 282340, 356802] ----------------------------------------------------------------- Proposition No. , 4.19, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r] and Goals Against Vector, [r + 4, r, r, r], is given by 4 3 2 (r - 1) (2 r + 27 r + 27 r - 118 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r-1)*(2*r^4+27*r^3+27*r^2-118*r+60) The first few terms, starting at, 1, are [0, 6, 56, 226, 636, 1458, 2924, 5334, 9064, 14574, 22416, 33242, 47812, 67002, 91812, 123374, 162960, 211990, 272040, 344850] If the Goals For Vector Was [r + 4, r, r, r] ----------------------------------------------------------------- Proposition No. , 4.20, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 1], is given by r (r + 9) (2 r + 1) (r + 2) (r + 1) ----------------------------------- 30 and in Maple input form 1/30*r*(r+9)*(2*r+1)*(r+2)*(r+1) The first few terms, starting at, 1, are [6, 44, 168, 468, 1078, 2184, 4032, 6936, 11286, 17556, 26312, 38220, 54054, 74704, 101184, 134640, 176358, 227772, 290472, 366212] ----------------------------------------------------------------- Proposition No. , 4.21, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r], is given by 4 3 2 r (2 r + 25 r + 40 r - 40 r + 3) ----------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+25*r^3+40*r^2-40*r+3) The first few terms, starting at, 1, are [1, 21, 108, 346, 863, 1839, 3514, 6196, 10269, 16201, 24552, 35982, 51259, 71267, 97014, 129640, 170425, 220797, 282340, 356802] ----------------------------------------------------------------- Proposition No. , 4.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r + 2, r + 2, r, r], is given by 5 4 3 2 14 1/15 r + 5/6 r + r - 5/6 r + -- r - 1 15 and in Maple input form 1/15*r^5+5/6*r^4+r^3-5/6*r^2+14/15*r-1 The first few terms, starting at, 1, are [1, 21, 105, 335, 837, 1789, 3429, 6063, 10073, 15925, 24177, 35487, 50621, 70461, 96013, 128415, 168945, 219029, 280249, 354351] ----------------------------------------------------------------- Proposition No. , 4.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r + 3, r + 1, r, r], is given by 4 3 2 (r - 1) (2 r + 27 r + 27 r - 118 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r-1)*(2*r^4+27*r^3+27*r^2-118*r+60) The first few terms, starting at, 1, are [0, 6, 56, 226, 636, 1458, 2924, 5334, 9064, 14574, 22416, 33242, 47812, 67002, 91812, 123374, 162960, 211990, 272040, 344850] ----------------------------------------------------------------- Proposition No. , 4.24, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r + 4, r, r, r], is given by 4 3 2 (r - 1) (2 r + 27 r - 23 r - 243 r + 360) -------------------------------------------- 30 and in Maple input form 1/30*(r-1)*(2*r^4+27*r^3-23*r^2-243*r+360) The first few terms, starting at, 1, are [0, 1, 21, 126, 426, 1083, 2319, 4424, 7764, 12789, 20041, 30162, 43902, 62127, 85827, 116124, 154280, 201705, 259965, 330790] This concludes this chapter, that took, 1.120, seconds to produce. ---------------------------------------------------- Chapter Number, 5 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 5 If the Goals For Vector Was [r + 2, r + 1, r + 1, r + 1] the smallest entry of, [2, 1, 1, 1], [2, 1, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 5.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 40) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+40) The first few terms, starting at, 1, are [30, 118, 338, 798, 1652, 3108, 5436, 8976, 14146, 21450, 31486, 44954, 62664, 85544, 114648, 151164, 196422, 251902, 319242, 400246] ----------------------------------------------------------------- Proposition No. , 5.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 10) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+10) The first few terms, starting at, 1, are [18, 88, 278, 693, 1484, 2856, 5076, 8481, 13486, 20592, 30394, 43589, 60984, 83504, 112200, 148257, 193002, 247912, 314622, 394933] ----------------------------------------------------------------- Proposition No. , 5.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 2, r, r], is given by 4 3 2 (r + 1) (4 r + 51 r + 169 r + 166 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+169*r^2+166*r+60) The first few terms, starting at, 1, are [15, 77, 252, 643, 1399, 2723, 4880, 8205, 13111, 20097, 29756, 42783, 59983, 82279, 110720, 146489, 190911, 245461, 311772, 391643] ----------------------------------------------------------------- Proposition No. , 5.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 4, r + 1, r, r], is given by 2 r (r + 3) (r + 1) (4 r + 39 r + 2) ----------------------------------- 60 and in Maple input form 1/60*r*(r+3)*(r+1)*(4*r^2+39*r+2) The first few terms, starting at, 1, are [6, 48, 186, 518, 1188, 2394, 4396, 7524, 12186, 18876, 28182, 40794, 57512, 79254, 107064, 142120, 185742, 239400, 304722, 383502] ----------------------------------------------------------------- Proposition No. , 5.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 5, r, r, r], is given by 4 3 2 r (4 r + 55 r + 80 r - 115 r + 36) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+80*r^2-115*r+36) The first few terms, starting at, 1, are [1, 21, 111, 360, 903, 1929, 3689, 6504, 10773, 16981, 25707, 37632, 53547, 74361, 101109, 134960, 177225, 229365, 292999, 369912] If the Goals For Vector Was [r + 2, r + 2, r + 1, r] ----------------------------------------------------------------- Proposition No. , 5.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 40) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+40) The first few terms, starting at, 1, are [30, 118, 338, 798, 1652, 3108, 5436, 8976, 14146, 21450, 31486, 44954, 62664, 85544, 114648, 151164, 196422, 251902, 319242, 400246] ----------------------------------------------------------------- Proposition No. , 5.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 93 r + 60) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+93*r+60) The first few terms, starting at, 1, are [20, 90, 278, 688, 1470, 2828, 5028, 8406, 13376, 20438, 30186, 43316, 60634, 83064, 111656, 147594, 192204, 246962, 313502, 393624] ----------------------------------------------------------------- Proposition No. , 5.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 73 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+73*r+30) The first few terms, starting at, 1, are [15, 76, 248, 633, 1379, 2688, 4824, 8121, 12991, 19932, 29536, 42497, 59619, 81824, 110160, 145809, 190095, 244492, 310632, 390313] ----------------------------------------------------------------- Proposition No. , 5.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 3, r + 2, r, r], is given by 4 3 2 (r + 1) (4 r + 51 r + 129 r + 146 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+129*r^2+146*r+60) The first few terms, starting at, 1, are [13, 67, 224, 583, 1289, 2541, 4600, 7797, 12541, 19327, 28744, 41483, 58345, 80249, 108240, 143497, 187341, 241243, 306832, 385903] ----------------------------------------------------------------- Proposition No. , 5.10, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 4, r + 1, r, r], is given by 3 2 r (r + 1) (4 r + 51 r + 89 r + 36) ------------------------------------ 60 and in Maple input form 1/60*r*(r+1)*(4*r^3+51*r^2+89*r+36) The first few terms, starting at, 1, are [6, 45, 174, 488, 1128, 2289, 4228, 7272, 11826, 18381, 27522, 39936, 56420, 77889, 105384, 140080, 183294, 236493, 301302, 379512] ----------------------------------------------------------------- Proposition No. , 5.11, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r + 5, r, r, r], is given by 5 11 4 3 17 2 43 1/15 r + -- r + r - -- r + -- r - 1 12 12 30 and in Maple input form 1/15*r^5+11/12*r^4+r^3-17/12*r^2+43/30*r-1 The first few terms, starting at, 1, are [1, 21, 108, 349, 877, 1879, 3604, 6371, 10577, 16705, 25332, 37137, 52909, 73555, 100108, 133735, 175745, 227597, 290908, 367461] If the Goals For Vector Was [r + 3, r + 1, r + 1, r] ----------------------------------------------------------------- Proposition No. , 5.12, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 10) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+10) The first few terms, starting at, 1, are [18, 88, 278, 693, 1484, 2856, 5076, 8481, 13486, 20592, 30394, 43589, 60984, 83504, 112200, 148257, 193002, 247912, 314622, 394933] ----------------------------------------------------------------- Proposition No. , 5.13, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 73 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+73*r+30) The first few terms, starting at, 1, are [15, 76, 248, 633, 1379, 2688, 4824, 8121, 12991, 19932, 29536, 42497, 59619, 81824, 110160, 145809, 190095, 244492, 310632, 390313] ----------------------------------------------------------------- Proposition No. , 5.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 2 r (r + 3) (r + 1) (4 r + 39 r + 2) ----------------------------------- 60 and in Maple input form 1/60*r*(r+3)*(r+1)*(4*r^2+39*r+2) The first few terms, starting at, 1, are [6, 48, 186, 518, 1188, 2394, 4396, 7524, 12186, 18876, 28182, 40794, 57512, 79254, 107064, 142120, 185742, 239400, 304722, 383502] ----------------------------------------------------------------- Proposition No. , 5.15, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 2, r, r], is given by 4 3 2 r (4 r + 55 r + 150 r + 95 r + 56) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+150*r^2+95*r+56) The first few terms, starting at, 1, are [6, 45, 175, 492, 1138, 2309, 4263, 7328, 11910, 18501, 27687, 40156, 56706, 78253, 105839, 140640, 183974, 237309, 302271, 380652] ----------------------------------------------------------------- Proposition No. , 5.16, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 4, r + 1, r, r], is given by 4 3 2 r (4 r + 55 r + 90 r - 145 r + 56) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+90*r^2-145*r+56) The first few terms, starting at, 1, are [1, 21, 112, 364, 913, 1949, 3724, 6560, 10857, 17101, 25872, 37852, 53833, 74725, 101564, 135520, 177905, 230181, 293968, 371052] ----------------------------------------------------------------- Proposition No. , 5.17, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r + 5, r, r, r], is given by 4 3 2 (r - 1) (4 r + 59 r + 49 r - 306 r + 240) -------------------------------------------- 60 and in Maple input form 1/60*(r-1)*(4*r^4+59*r^3+49*r^2-306*r+240) The first few terms, starting at, 1, are [0, 6, 56, 230, 654, 1508, 3034, 5544, 9428, 15162, 23316, 34562, 49682, 69576, 95270, 127924, 168840, 219470, 281424, 356478] If the Goals For Vector Was [r + 3, r + 2, r, r] ----------------------------------------------------------------- Proposition No. , 5.18, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r + 1], is given by 4 3 2 (r + 1) (4 r + 51 r + 169 r + 166 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+169*r^2+166*r+60) The first few terms, starting at, 1, are [15, 77, 252, 643, 1399, 2723, 4880, 8205, 13111, 20097, 29756, 42783, 59983, 82279, 110720, 146489, 190911, 245461, 311772, 391643] ----------------------------------------------------------------- Proposition No. , 5.19, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 4 3 2 (r + 1) (4 r + 51 r + 129 r + 146 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+129*r^2+146*r+60) The first few terms, starting at, 1, are [13, 67, 224, 583, 1289, 2541, 4600, 7797, 12541, 19327, 28744, 41483, 58345, 80249, 108240, 143497, 187341, 241243, 306832, 385903] ----------------------------------------------------------------- Proposition No. , 5.20, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 4 3 2 r (4 r + 55 r + 150 r + 95 r + 56) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+150*r^2+95*r+56) The first few terms, starting at, 1, are [6, 45, 175, 492, 1138, 2309, 4263, 7328, 11910, 18501, 27687, 40156, 56706, 78253, 105839, 140640, 183974, 237309, 302271, 380652] ----------------------------------------------------------------- Proposition No. , 5.21, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r + 3, r + 2, r, r], is given by 3 2 r (r + 1) (4 r + 51 r + 59 r + 66) ------------------------------------ 60 and in Maple input form 1/60*r*(r+1)*(4*r^3+51*r^2+59*r+66) The first few terms, starting at, 1, are [6, 42, 162, 458, 1068, 2184, 4060, 7020, 11466, 17886, 26862, 39078, 55328, 76524, 103704, 138040, 180846, 233586, 297882, 375522] ----------------------------------------------------------------- Proposition No. , 5.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r + 4, r + 1, r, r], is given by 5 11 4 3 17 2 43 1/15 r + -- r + r - -- r + -- r - 1 12 12 30 and in Maple input form 1/15*r^5+11/12*r^4+r^3-17/12*r^2+43/30*r-1 The first few terms, starting at, 1, are [1, 21, 108, 349, 877, 1879, 3604, 6371, 10577, 16705, 25332, 37137, 52909, 73555, 100108, 133735, 175745, 227597, 290908, 367461] If the Goals For Vector Was [r + 4, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 5.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r + 1], is given by 2 r (r + 3) (r + 1) (4 r + 39 r + 2) ----------------------------------- 60 and in Maple input form 1/60*r*(r+3)*(r+1)*(4*r^2+39*r+2) The first few terms, starting at, 1, are [6, 48, 186, 518, 1188, 2394, 4396, 7524, 12186, 18876, 28182, 40794, 57512, 79254, 107064, 142120, 185742, 239400, 304722, 383502] ----------------------------------------------------------------- Proposition No. , 5.24, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 3 2 r (r + 1) (4 r + 51 r + 89 r + 36) ------------------------------------ 60 and in Maple input form 1/60*r*(r+1)*(4*r^3+51*r^2+89*r+36) The first few terms, starting at, 1, are [6, 45, 174, 488, 1128, 2289, 4228, 7272, 11826, 18381, 27522, 39936, 56420, 77889, 105384, 140080, 183294, 236493, 301302, 379512] ----------------------------------------------------------------- Proposition No. , 5.25, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 4 3 2 r (4 r + 55 r + 90 r - 145 r + 56) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+90*r^2-145*r+56) The first few terms, starting at, 1, are [1, 21, 112, 364, 913, 1949, 3724, 6560, 10857, 17101, 25872, 37852, 53833, 74725, 101564, 135520, 177905, 230181, 293968, 371052] ----------------------------------------------------------------- Proposition No. , 5.26, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r + 3, r + 2, r, r], is given by 5 11 4 3 17 2 43 1/15 r + -- r + r - -- r + -- r - 1 12 12 30 and in Maple input form 1/15*r^5+11/12*r^4+r^3-17/12*r^2+43/30*r-1 The first few terms, starting at, 1, are [1, 21, 108, 349, 877, 1879, 3604, 6371, 10577, 16705, 25332, 37137, 52909, 73555, 100108, 133735, 175745, 227597, 290908, 367461] ----------------------------------------------------------------- Proposition No. , 5.27, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r + 4, r + 1, r, r], is given by 4 3 2 (r - 1) (4 r + 59 r + 49 r - 306 r + 240) -------------------------------------------- 60 and in Maple input form 1/60*(r-1)*(4*r^4+59*r^3+49*r^2-306*r+240) The first few terms, starting at, 1, are [0, 6, 56, 230, 654, 1508, 3034, 5544, 9428, 15162, 23316, 34562, 49682, 69576, 95270, 127924, 168840, 219470, 281424, 356478] If the Goals For Vector Was [r + 5, r, r, r] ----------------------------------------------------------------- Proposition No. , 5.28, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r + 2, r + 1, r + 1, r + 1], is given by 4 3 2 r (4 r + 55 r + 80 r - 115 r + 36) ------------------------------------- 60 and in Maple input form 1/60*r*(4*r^4+55*r^3+80*r^2-115*r+36) The first few terms, starting at, 1, are [1, 21, 111, 360, 903, 1929, 3689, 6504, 10773, 16981, 25707, 37632, 53547, 74361, 101109, 134960, 177225, 229365, 292999, 369912] ----------------------------------------------------------------- Proposition No. , 5.29, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r], is given by 5 11 4 3 17 2 43 1/15 r + -- r + r - -- r + -- r - 1 12 12 30 and in Maple input form 1/15*r^5+11/12*r^4+r^3-17/12*r^2+43/30*r-1 The first few terms, starting at, 1, are [1, 21, 108, 349, 877, 1879, 3604, 6371, 10577, 16705, 25332, 37137, 52909, 73555, 100108, 133735, 175745, 227597, 290908, 367461] ----------------------------------------------------------------- Proposition No. , 5.30, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r], is given by 4 3 2 (r - 1) (4 r + 59 r + 49 r - 306 r + 240) -------------------------------------------- 60 and in Maple input form 1/60*(r-1)*(4*r^4+59*r^3+49*r^2-306*r+240) The first few terms, starting at, 1, are [0, 6, 56, 230, 654, 1508, 3034, 5544, 9428, 15162, 23316, 34562, 49682, 69576, 95270, 127924, 168840, 219470, 281424, 356478] This concludes this chapter, that took, 4.109, seconds to produce. ---------------------------------------------------- Chapter Number, 6 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 4, Teams with many possible Goals For and Goals Against Ouctomes, Where the Relative \ Difference to All Scoring the same totalled, 6 If the Goals For Vector Was [r + 2, r + 2, r + 1, r + 1] the smallest entry of, [2, 2, 1, 1], [2, 2, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 6.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 15) --------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+15) The first few terms, starting at, 1, are [40, 148, 408, 938, 1904, 3528, 6096, 9966, 15576, 23452, 34216, 48594, 67424, 91664, 122400, 160854, 208392, 266532, 336952, 421498] the smallest entry of, [2, 2, 1, 1], [3, 1, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 6.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 69 r + 60) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+69*r+60) The first few terms, starting at, 1, are [31, 124, 358, 848, 1757, 3304, 5772, 9516, 14971, 22660, 33202, 47320, 65849, 89744, 120088, 158100, 205143, 262732, 332542, 416416] ----------------------------------------------------------------- Proposition No. , 6.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 22 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+22*r+15) The first few terms, starting at, 1, are [20, 92, 288, 718, 1540, 2968, 5280, 8826, 14036, 21428, 31616, 45318, 63364, 86704, 116416, 153714, 199956, 256652, 325472, 408254] ----------------------------------------------------------------- Proposition No. , 6.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 49 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+49*r+15) The first few terms, starting at, 1, are [18, 90, 288, 723, 1554, 2996, 5328, 8901, 14146, 21582, 31824, 45591, 63714, 87144, 116960, 154377, 200754, 257602, 326592, 409563] ----------------------------------------------------------------- Proposition No. , 6.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 2 r (r + 2) (r + 1) (r + 12 r + 2) --------------------------------- 15 and in Maple input form 1/15*r*(r+2)*(r+1)*(r^2+12*r+2) The first few terms, starting at, 1, are [6, 48, 188, 528, 1218, 2464, 4536, 7776, 12606, 19536, 29172, 42224, 59514, 81984, 110704, 146880, 191862, 247152, 314412, 395472] ----------------------------------------------------------------- Proposition No. , 6.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r + 6, r, r, r], is given by 5 4 3 2 29 1/15 r + r + r - 2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+r^3-2*r^2+29/15*r-1 The first few terms, starting at, 1, are [1, 21, 111, 363, 917, 1969, 3779, 6679, 11081, 17485, 26487, 38787, 55197, 76649, 104203, 139055, 182545, 236165, 301567, 380571] If the Goals For Vector Was [r + 2, r + 2, r + 2, r] ----------------------------------------------------------------- Proposition No. , 6.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 15) --------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+15) The first few terms, starting at, 1, are [40, 148, 408, 938, 1904, 3528, 6096, 9966, 15576, 23452, 34216, 48594, 67424, 91664, 122400, 160854, 208392, 266532, 336952, 421498] ----------------------------------------------------------------- Proposition No. , 6.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 64 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+64*r+45) The first few terms, starting at, 1, are [27, 114, 338, 813, 1701, 3220, 5652, 9351, 14751, 22374, 32838, 46865, 65289, 89064, 119272, 157131, 204003, 261402, 331002, 414645] ----------------------------------------------------------------- Proposition No. , 6.10, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 10) --------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+10) The first few terms, starting at, 1, are [32, 128, 368, 868, 1792, 3360, 5856, 9636, 15136, 22880, 33488, 47684, 66304, 90304, 120768, 158916, 206112, 263872, 333872, 417956] ----------------------------------------------------------------- Proposition No. , 6.11, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 27 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+27*r+15) The first few terms, starting at, 1, are [22, 100, 308, 758, 1610, 3080, 5448, 9066, 14366, 21868, 32188, 46046, 64274, 87824, 117776, 155346, 201894, 258932, 328132, 411334] ----------------------------------------------------------------- Proposition No. , 6.12, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.13, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 24 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+24*r+15) The first few terms, starting at, 1, are [13, 70, 238, 623, 1379, 2716, 4908, 8301, 13321, 20482, 30394, 43771, 61439, 84344, 113560, 150297, 195909, 251902, 319942, 401863] ----------------------------------------------------------------- Proposition No. , 6.15, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 4 3 2 r (2 r + 30 r + 70 r + 45 r + 33) ------------------------------------ 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+70*r^2+45*r+33) The first few terms, starting at, 1, are [6, 45, 177, 502, 1168, 2379, 4403, 7580, 12330, 19161, 28677, 41586, 58708, 80983, 109479, 145400, 190094, 245061, 311961, 392622] ----------------------------------------------------------------- Proposition No. , 6.16, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 6, r, r, r], is given by 5 4 3 2 83 1/15 r + r + 2/3 r - 3/2 r + -- r - 2 30 and in Maple input form 1/15*r^5+r^4+2/3*r^3-3/2*r^2+83/30*r-2 The first few terms, starting at, 1, are [1, 21, 108, 352, 891, 1919, 3694, 6546, 10885, 17209, 26112, 38292, 54559, 75843, 103202, 137830, 181065, 234397, 299476, 378120] If the Goals For Vector Was [r + 3, r + 1, r + 1, r + 1] the smallest entry of, [3, 1, 1, 1], [2, 2, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 6.17, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 10) --------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+10) The first few terms, starting at, 1, are [32, 128, 368, 868, 1792, 3360, 5856, 9636, 15136, 22880, 33488, 47684, 66304, 90304, 120768, 158916, 206112, 263872, 333872, 417956] the smallest entry of, [3, 1, 1, 1], [3, 1, 1, 1], should have been 0 Bad input ----------------------------------------------------------------- Proposition No. , 6.18, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 4 3 2 (r + 1) (r + 14 r + 51 r + 54 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+51*r^2+54*r+15) The first few terms, starting at, 1, are [18, 91, 292, 733, 1574, 3031, 5384, 8985, 14266, 21747, 32044, 45877, 64078, 87599, 117520, 155057, 201570, 258571, 327732, 410893] ----------------------------------------------------------------- Proposition No. , 6.19, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 5 4 3 2 64 1/15 r + r + 11/3 r + 5 r + -- r + 1 15 and in Maple input form 1/15*r^5+r^4+11/3*r^3+5*r^2+64/15*r+1 The first few terms, starting at, 1, are [15, 77, 255, 657, 1439, 2813, 5055, 8513, 13615, 20877, 30911, 44433, 62271, 85373, 114815, 151809, 197711, 254029, 322431, 404753] ----------------------------------------------------------------- Proposition No. , 6.20, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 3 2 r (r + 1) (r + 14 r + 36 r - 6) --------------------------------- 15 and in Maple input form 1/15*r*(r+1)*(r^3+14*r^2+36*r-6) The first few terms, starting at, 1, are [6, 52, 204, 568, 1298, 2604, 4760, 8112, 13086, 20196, 30052, 43368, 60970, 83804, 112944, 149600, 195126, 251028, 318972, 400792] ----------------------------------------------------------------- Proposition No. , 6.21, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 4 3 2 r (r + 15 r + 45 r + 15 r + 14) ---------------------------------- 15 and in Maple input form 1/15*r*(r^4+15*r^3+45*r^2+15*r+14) The first few terms, starting at, 1, are [6, 48, 190, 536, 1238, 2504, 4606, 7888, 12774, 19776, 29502, 42664, 60086, 82712, 111614, 148000, 193222, 248784, 316350, 397752] ----------------------------------------------------------------- Proposition No. , 6.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 4 3 2 r (2 r + 30 r + 50 r - 105 r + 53) ------------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+50*r^2-105*r+53) The first few terms, starting at, 1, are [1, 21, 116, 382, 963, 2059, 3934, 6924, 11445, 18001, 27192, 39722, 56407, 78183, 106114, 141400, 185385, 239565, 305596, 385302] ----------------------------------------------------------------- Proposition No. , 6.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r + 6, r, r, r], is given by 4 3 2 (r - 1) (r + 16 r + 11 r - 94 r + 90) ---------------------------------------- 15 and in Maple input form 1/15*(r-1)*(r^4+16*r^3+11*r^2-94*r+90) The first few terms, starting at, 1, are [0, 6, 56, 234, 672, 1558, 3144, 5754, 9792, 15750, 24216, 35882, 51552, 72150, 98728, 132474, 174720, 226950, 290808, 368106] If the Goals For Vector Was [r + 3, r + 2, r + 1, r] ----------------------------------------------------------------- Proposition No. , 6.24, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 69 r + 60) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+69*r+60) The first few terms, starting at, 1, are [31, 124, 358, 848, 1757, 3304, 5772, 9516, 14971, 22660, 33202, 47320, 65849, 89744, 120088, 158100, 205143, 262732, 332542, 416416] ----------------------------------------------------------------- Proposition No. , 6.25, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 27 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+27*r+15) The first few terms, starting at, 1, are [22, 100, 308, 758, 1610, 3080, 5448, 9066, 14366, 21868, 32188, 46046, 64274, 87824, 117776, 155346, 201894, 258932, 328132, 411334] ----------------------------------------------------------------- Proposition No. , 6.26, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 4 3 2 (r + 1) (r + 14 r + 51 r + 54 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+51*r^2+54*r+15) The first few terms, starting at, 1, are [18, 91, 292, 733, 1574, 3031, 5384, 8985, 14266, 21747, 32044, 45877, 64078, 87599, 117520, 155057, 201570, 258571, 327732, 410893] ----------------------------------------------------------------- Proposition No. , 6.27, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.28, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 4 3 2 (r + 1) (2 r + 28 r + 57 r + 78 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+57*r^2+78*r+30) The first few terms, starting at, 1, are [13, 67, 226, 593, 1319, 2611, 4740, 8049, 12961, 19987, 29734, 42913, 60347, 82979, 111880, 148257, 193461, 248995, 316522, 397873] ----------------------------------------------------------------- Proposition No. , 6.29, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 3 2 r (r + 3) (2 r + 24 r + 13 r + 6) ----------------------------------- 30 and in Maple input form 1/30*r*(r+3)*(2*r^3+24*r^2+13*r+6) The first few terms, starting at, 1, are [6, 48, 189, 532, 1228, 2484, 4571, 7832, 12690, 19656, 29337, 42444, 59800, 82348, 111159, 147440, 192542, 247968, 315381, 396612] ----------------------------------------------------------------- Proposition No. , 6.30, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 4 3 2 r (2 r + 30 r + 70 r + 45 r + 33) ------------------------------------ 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+70*r^2+45*r+33) The first few terms, starting at, 1, are [6, 45, 177, 502, 1168, 2379, 4403, 7580, 12330, 19161, 28677, 41586, 58708, 80983, 109479, 145400, 190094, 245061, 311961, 392622] ----------------------------------------------------------------- Proposition No. , 6.31, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 5 4 3 2 34 1/15 r + r + 7/6 r - 5/2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+7/6*r^3-5/2*r^2+34/15*r-1 The first few terms, starting at, 1, are [1, 21, 112, 367, 927, 1989, 3814, 6735, 11165, 17605, 26652, 39007, 55483, 77013, 104658, 139615, 183225, 236981, 302536, 381711] If the Goals For Vector Was [r + 3, r + 3, r, r] ----------------------------------------------------------------- Proposition No. , 6.32, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 22 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+22*r+15) The first few terms, starting at, 1, are [20, 92, 288, 718, 1540, 2968, 5280, 8826, 14036, 21428, 31616, 45318, 63364, 86704, 116416, 153714, 199956, 256652, 325472, 408254] ----------------------------------------------------------------- Proposition No. , 6.33, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.34, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 5 4 3 2 64 1/15 r + r + 11/3 r + 5 r + -- r + 1 15 and in Maple input form 1/15*r^5+r^4+11/3*r^3+5*r^2+64/15*r+1 The first few terms, starting at, 1, are [15, 77, 255, 657, 1439, 2813, 5055, 8513, 13615, 20877, 30911, 44433, 62271, 85373, 114815, 151809, 197711, 254029, 322431, 404753] ----------------------------------------------------------------- Proposition No. , 6.35, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 4 3 2 (r + 1) (2 r + 28 r + 57 r + 78 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+57*r^2+78*r+30) The first few terms, starting at, 1, are [13, 67, 226, 593, 1319, 2611, 4740, 8049, 12961, 19987, 29734, 42913, 60347, 82979, 111880, 148257, 193461, 248995, 316522, 397873] ----------------------------------------------------------------- Proposition No. , 6.36, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 5 4 3 2 83 1/15 r + r + 5/3 r + 9/2 r + -- r + 1 30 and in Maple input form 1/15*r^5+r^4+5/3*r^3+9/2*r^2+83/30*r+1 The first few terms, starting at, 1, are [11, 56, 192, 515, 1169, 2354, 4334, 7445, 12103, 18812, 28172, 40887, 57773, 79766, 107930, 143465, 187715, 242176, 308504, 388523] ----------------------------------------------------------------- Proposition No. , 6.37, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 5 4 3 2 34 1/15 r + r + 13/6 r + 3/2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+13/6*r^3+3/2*r^2+34/15*r-1 The first few terms, starting at, 1, are [6, 45, 175, 495, 1152, 2349, 4353, 7503, 12218, 19005, 28467, 41311, 58356, 80541, 108933, 144735, 189294, 244109, 310839, 391311] ----------------------------------------------------------------- Proposition No. , 6.38, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 2 2 r (r + r + 3) (r + 14 r + 3) ------------------------------ 15 and in Maple input form 1/15*r*(r^2+r+3)*(r^2+14*r+3) The first few terms, starting at, 1, are [6, 42, 162, 460, 1078, 2214, 4130, 7160, 11718, 18306, 27522, 40068, 56758, 78526, 106434, 141680, 185606, 239706, 305634, 385212] If the Goals For Vector Was [r + 4, r + 1, r + 1, r] ----------------------------------------------------------------- Proposition No. , 6.39, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 49 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+49*r+15) The first few terms, starting at, 1, are [18, 90, 288, 723, 1554, 2996, 5328, 8901, 14146, 21582, 31824, 45591, 63714, 87144, 116960, 154377, 200754, 257602, 326592, 409563] ----------------------------------------------------------------- Proposition No. , 6.40, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.41, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 3 2 r (r + 1) (r + 14 r + 36 r - 6) --------------------------------- 15 and in Maple input form 1/15*r*(r+1)*(r^3+14*r^2+36*r-6) The first few terms, starting at, 1, are [6, 52, 204, 568, 1298, 2604, 4760, 8112, 13086, 20196, 30052, 43368, 60970, 83804, 112944, 149600, 195126, 251028, 318972, 400792] ----------------------------------------------------------------- Proposition No. , 6.42, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 3 2 r (r + 3) (2 r + 24 r + 13 r + 6) ----------------------------------- 30 and in Maple input form 1/30*r*(r+3)*(2*r^3+24*r^2+13*r+6) The first few terms, starting at, 1, are [6, 48, 189, 532, 1228, 2484, 4571, 7832, 12690, 19656, 29337, 42444, 59800, 82348, 111159, 147440, 192542, 247968, 315381, 396612] ----------------------------------------------------------------- Proposition No. , 6.43, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 5 4 3 2 34 1/15 r + r + 13/6 r + 3/2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+13/6*r^3+3/2*r^2+34/15*r-1 The first few terms, starting at, 1, are [6, 45, 175, 495, 1152, 2349, 4353, 7503, 12218, 19005, 28467, 41311, 58356, 80541, 108933, 144735, 189294, 244109, 310839, 391311] ----------------------------------------------------------------- Proposition No. , 6.44, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 4 3 2 r (2 r + 30 r + 50 r - 105 r + 53) ------------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+50*r^2-105*r+53) The first few terms, starting at, 1, are [1, 21, 116, 382, 963, 2059, 3934, 6924, 11445, 18001, 27192, 39722, 56407, 78183, 106114, 141400, 185385, 239565, 305596, 385302] ----------------------------------------------------------------- Proposition No. , 6.45, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 5 4 3 2 41 1/15 r + r + 4/3 r - 7/2 r + -- r - 2 10 and in Maple input form 1/15*r^5+r^4+4/3*r^3-7/2*r^2+41/10*r-2 The first few terms, starting at, 1, are [1, 21, 112, 368, 931, 1999, 3834, 6770, 11221, 17689, 26772, 39172, 55703, 77299, 105022, 140070, 183785, 237661, 303352, 382680] ----------------------------------------------------------------- Proposition No. , 6.46, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 4 3 2 (r - 1) (2 r + 32 r + 27 r - 213 r + 210) -------------------------------------------- 30 and in Maple input form 1/30*(r-1)*(2*r^4+32*r^3+27*r^2-213*r+210) The first few terms, starting at, 1, are [0, 6, 56, 235, 676, 1568, 3164, 5789, 9848, 15834, 24336, 36047, 51772, 72436, 99092, 132929, 175280, 227630, 291624, 369075] If the Goals For Vector Was [r + 4, r + 2, r, r] ----------------------------------------------------------------- Proposition No. , 6.47, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 34 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+34*r+15) The first few terms, starting at, 1, are [15, 78, 258, 663, 1449, 2828, 5076, 8541, 13651, 20922, 30966, 44499, 62349, 85464, 114920, 151929, 197847, 254182, 322602, 404943] ----------------------------------------------------------------- Proposition No. , 6.48, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 24 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+24*r+15) The first few terms, starting at, 1, are [13, 70, 238, 623, 1379, 2716, 4908, 8301, 13321, 20482, 30394, 43771, 61439, 84344, 113560, 150297, 195909, 251902, 319942, 401863] ----------------------------------------------------------------- Proposition No. , 6.49, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 4 3 2 r (r + 15 r + 45 r + 15 r + 14) ---------------------------------- 15 and in Maple input form 1/15*r*(r^4+15*r^3+45*r^2+15*r+14) The first few terms, starting at, 1, are [6, 48, 190, 536, 1238, 2504, 4606, 7888, 12774, 19776, 29502, 42664, 60086, 82712, 111614, 148000, 193222, 248784, 316350, 397752] ----------------------------------------------------------------- Proposition No. , 6.50, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 4 3 2 r (2 r + 30 r + 70 r + 45 r + 33) ------------------------------------ 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+70*r^2+45*r+33) The first few terms, starting at, 1, are [6, 45, 177, 502, 1168, 2379, 4403, 7580, 12330, 19161, 28677, 41586, 58708, 80983, 109479, 145400, 190094, 245061, 311961, 392622] ----------------------------------------------------------------- Proposition No. , 6.51, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 3, r + 3, r, r], is given by 2 2 r (r + r + 3) (r + 14 r + 3) ------------------------------ 15 and in Maple input form 1/15*r*(r^2+r+3)*(r^2+14*r+3) The first few terms, starting at, 1, are [6, 42, 162, 460, 1078, 2214, 4130, 7160, 11718, 18306, 27522, 40068, 56758, 78526, 106434, 141680, 185606, 239706, 305634, 385212] ----------------------------------------------------------------- Proposition No. , 6.52, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 5 4 3 2 41 1/15 r + r + 4/3 r - 7/2 r + -- r - 2 10 and in Maple input form 1/15*r^5+r^4+4/3*r^3-7/2*r^2+41/10*r-2 The first few terms, starting at, 1, are [1, 21, 112, 368, 931, 1999, 3834, 6770, 11221, 17689, 26772, 39172, 55703, 77299, 105022, 140070, 183785, 237661, 303352, 382680] ----------------------------------------------------------------- Proposition No. , 6.53, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r + 4, r + 2, r, r], is given by 5 4 3 2 83 1/15 r + r + 2/3 r - 3/2 r + -- r - 2 30 and in Maple input form 1/15*r^5+r^4+2/3*r^3-3/2*r^2+83/30*r-2 The first few terms, starting at, 1, are [1, 21, 108, 352, 891, 1919, 3694, 6546, 10885, 17209, 26112, 38292, 54559, 75843, 103202, 137830, 181065, 234397, 299476, 378120] If the Goals For Vector Was [r + 5, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 6.54, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 2 r (r + 2) (r + 1) (r + 12 r + 2) --------------------------------- 15 and in Maple input form 1/15*r*(r+2)*(r+1)*(r^2+12*r+2) The first few terms, starting at, 1, are [6, 48, 188, 528, 1218, 2464, 4536, 7776, 12606, 19536, 29172, 42224, 59514, 81984, 110704, 146880, 191862, 247152, 314412, 395472] ----------------------------------------------------------------- Proposition No. , 6.55, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 4 3 2 r (2 r + 30 r + 70 r + 45 r + 33) ------------------------------------ 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+70*r^2+45*r+33) The first few terms, starting at, 1, are [6, 45, 177, 502, 1168, 2379, 4403, 7580, 12330, 19161, 28677, 41586, 58708, 80983, 109479, 145400, 190094, 245061, 311961, 392622] ----------------------------------------------------------------- Proposition No. , 6.56, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 4 3 2 r (2 r + 30 r + 50 r - 105 r + 53) ------------------------------------- 30 and in Maple input form 1/30*r*(2*r^4+30*r^3+50*r^2-105*r+53) The first few terms, starting at, 1, are [1, 21, 116, 382, 963, 2059, 3934, 6924, 11445, 18001, 27192, 39722, 56407, 78183, 106114, 141400, 185385, 239565, 305596, 385302] ----------------------------------------------------------------- Proposition No. , 6.57, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 3, r + 2, r + 1, r], is given by 5 4 3 2 34 1/15 r + r + 7/6 r - 5/2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+7/6*r^3-5/2*r^2+34/15*r-1 The first few terms, starting at, 1, are [1, 21, 112, 367, 927, 1989, 3814, 6735, 11165, 17605, 26652, 39007, 55483, 77013, 104658, 139615, 183225, 236981, 302536, 381711] ----------------------------------------------------------------- Proposition No. , 6.58, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 4, r + 1, r + 1, r], is given by 4 3 2 (r - 1) (2 r + 32 r + 27 r - 213 r + 210) -------------------------------------------- 30 and in Maple input form 1/30*(r-1)*(2*r^4+32*r^3+27*r^2-213*r+210) The first few terms, starting at, 1, are [0, 6, 56, 235, 676, 1568, 3164, 5789, 9848, 15834, 24336, 36047, 51772, 72436, 99092, 132929, 175280, 227630, 291624, 369075] ----------------------------------------------------------------- Proposition No. , 6.59, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 6, r, r, r], is given by 5 4 3 2 (r - 1) (r - 2) (r - 39 r + 539 r - 6705 r + 25056 r - 30240) - ----------------------------------------------------------------- 1260 and in Maple input form -1/1260*(r-1)*(r-2)*(r^5-39*r^4+539*r^3-6705*r^2+25056*r-30240) The first few terms, starting at, 1, are [0, 0, 6, 56, 252, 756, 1794, 3664, 6744, 11496, 18462, 28248, 41492, 58812, 80730, 107568, 139312, 175440, 214710, 254904] If the Goals For Vector Was [r + 6, r, r, r] ----------------------------------------------------------------- Proposition No. , 6.60, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 2, r + 2, r + 1, r + 1], is given by 5 4 3 2 29 1/15 r + r + r - 2 r + -- r - 1 15 and in Maple input form 1/15*r^5+r^4+r^3-2*r^2+29/15*r-1 The first few terms, starting at, 1, are [1, 21, 111, 363, 917, 1969, 3779, 6679, 11081, 17485, 26487, 38787, 55197, 76649, 104203, 139055, 182545, 236165, 301567, 380571] ----------------------------------------------------------------- Proposition No. , 6.61, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 2, r + 2, r + 2, r], is given by 5 4 3 2 83 1/15 r + r + 2/3 r - 3/2 r + -- r - 2 30 and in Maple input form 1/15*r^5+r^4+2/3*r^3-3/2*r^2+83/30*r-2 The first few terms, starting at, 1, are [1, 21, 108, 352, 891, 1919, 3694, 6546, 10885, 17209, 26112, 38292, 54559, 75843, 103202, 137830, 181065, 234397, 299476, 378120] ----------------------------------------------------------------- Proposition No. , 6.62, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 3, r + 1, r + 1, r + 1], is given by 4 3 2 (r - 1) (r + 16 r + 11 r - 94 r + 90) ---------------------------------------- 15 and in Maple input form 1/15*(r-1)*(r^4+16*r^3+11*r^2-94*r+90) The first few terms, starting at, 1, are [0, 6, 56, 234, 672, 1558, 3144, 5754, 9792, 15750, 24216, 35882, 51552, 72150, 98728, 132474, 174720, 226950, 290808, 368106] ----------------------------------------------------------------- Proposition No. , 6.63, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 5, r + 1, r, r], is given by 5 4 3 2 (r - 1) (r - 2) (r - 39 r + 539 r - 6705 r + 25056 r - 30240) - ----------------------------------------------------------------- 1260 and in Maple input form -1/1260*(r-1)*(r-2)*(r^5-39*r^4+539*r^3-6705*r^2+25056*r-30240) The first few terms, starting at, 1, are [0, 0, 6, 56, 252, 756, 1794, 3664, 6744, 11496, 18462, 28248, 41492, 58812, 80730, 107568, 139312, 175440, 214710, 254904] This concludes this chapter, that took, 8.995, seconds to produce. 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