Polynomial Expressions for the Number of Round-Robin Tournaments with , 4, Teams, 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, r, r, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 11 r + 10) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+11*r+10) The first few terms, starting at, 1, are [10, 48, 158, 413, 924, 1848, 3396, 5841, 9526, 14872, 22386, 32669, 46424, 64464, 87720, 117249, 154242, 200032, 256102, 324093] This concludes this chapter, that took, 0.101, 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, r, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (r + 4 r + 5) -------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+4*r+5) The first few terms, starting at, 1, are [16, 68, 208, 518, 1120, 2184, 3936, 6666, 10736, 16588, 24752, 35854, 50624, 69904, 94656, 125970, 165072, 213332, 272272, 343574] ----------------------------------------------------------------- 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, r, r, r + 2], 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] 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, r, r + 1, r + 1], 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. , 2.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r, r, r] and Goals Against Vector, [r, r, r, r + 2], is given by 4 3 2 (r + 1) (r + 9 r + 21 r + 29 r + 15) --------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+9*r^3+21*r^2+29*r+15) The first few terms, starting at, 1, are [10, 49, 164, 433, 974, 1953, 3592, 6177, 10066, 15697, 23596, 34385, 48790, 67649, 91920, 122689, 161178, 208753, 266932, 337393] This concludes this chapter, that took, 0.285, 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, r + 1, r + 1, r + 1], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 21 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+21*r+30) The first few terms, starting at, 1, are [22, 88, 258, 623, 1316, 2520, 4476, 7491, 11946, 18304, 27118, 39039, 54824, 75344, 101592, 134691, 175902, 226632, 288442, 363055] ----------------------------------------------------------------- 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, r, r + 1, r + 2], 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. , 3.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r, r, r + 3], is given by 3 2 (r + 2) (r + 1) (4 r + 33 r + 53 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+33*r^2+53*r+30) The first few terms, starting at, 1, are [12, 60, 198, 513, 1134, 2240, 4068, 6921, 11176, 17292, 25818, 37401, 52794, 72864, 98600, 131121, 171684, 221692, 282702, 356433] 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, r + 1, r + 1, r + 1], 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. , 3.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r] and Goals Against Vector, [r, r, r + 1, r + 2], is given by 3 2 (r + 2) (r + 1) (4 r + 33 r + 73 r + 60) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+33*r^2+73*r+60) The first few terms, starting at, 1, are [17, 74, 228, 568, 1225, 2380, 4272, 7206, 11561, 17798, 26468, 38220, 53809, 74104, 100096, 132906, 173793, 224162, 285572, 359744] ----------------------------------------------------------------- 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, r, r, r + 3], is given by 4 3 2 (r + 1) (4 r + 41 r + 99 r + 126 r + 60) ------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+41*r^3+99*r^2+126*r+60) The first few terms, starting at, 1, are [11, 55, 184, 483, 1079, 2149, 3928, 6717, 10891, 16907, 25312, 36751, 51975, 71849, 97360, 129625, 169899, 219583, 280232, 353563] 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, r + 1, r + 1, r + 1], is given by 3 2 (r + 2) (r + 1) (4 r + 33 r + 53 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+33*r^2+53*r+30) The first few terms, starting at, 1, are [12, 60, 198, 513, 1134, 2240, 4068, 6921, 11176, 17292, 25818, 37401, 52794, 72864, 98600, 131121, 171684, 221692, 282702, 356433] ----------------------------------------------------------------- Proposition No. , 3.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 2], is given by 4 3 2 (r + 1) (4 r + 41 r + 99 r + 126 r + 60) ------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+41*r^3+99*r^2+126*r+60) The first few terms, starting at, 1, are [11, 55, 184, 483, 1079, 2149, 3928, 6717, 10891, 16907, 25312, 36751, 51975, 71849, 97360, 129625, 169899, 219583, 280232, 353563] ----------------------------------------------------------------- Proposition No. , 3.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r] and Goals Against Vector, [r, r, r, r + 3], is given by 5 4 3 2 83 1/15 r + 3/4 r + 5/3 r + 15/4 r + -- r + 1 30 and in Maple input form 1/15*r^5+3/4*r^4+5/3*r^3+15/4*r^2+83/30*r+1 The first few terms, starting at, 1, are [10, 49, 165, 439, 994, 2003, 3697, 6373, 10402, 16237, 24421, 35595, 50506, 70015, 95105, 126889, 166618, 215689, 275653, 348223] This concludes this chapter, that took, 0.547, 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, r + 1, r + 1, r + 2], 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, r, r + 2, r + 2], 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, r, r + 1, r + 3], 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, r, r, r + 4], 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, r + 1, r + 1, r + 2], 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. , 4.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 2, r + 2], 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. , 4.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 1, r + 3], is given by 2 (2 r + 3) (r + 2) (r + 1) (r + 8 r + 10) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+3)*(r+2)*(r+1)*(r^2+8*r+10) The first few terms, starting at, 1, are [19, 84, 258, 638, 1365, 2632, 4692, 7866, 12551, 19228, 28470, 40950, 57449, 78864, 106216, 140658, 183483, 236132, 300202, 377454] ----------------------------------------------------------------- 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, r, r, r + 4], is given by 2 2 (r + 1) (r + 2 r + 2) (2 r + 19 r + 15) ----------------------------------------- 30 and in Maple input form 1/30*(r+1)*(r^2+2*r+2)*(2*r^2+19*r+15) The first few terms, starting at, 1, are [12, 61, 204, 533, 1184, 2345, 4264, 7257, 11716, 18117, 27028, 39117, 55160, 76049, 102800, 136561, 178620, 230413, 293532, 369733] 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, r + 1, r + 1, r + 2], 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. , 4.12, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r] and Goals Against Vector, [r, r, r + 2, r + 2], is given by 3 2 (r + 2) (r + 1) (2 r + 19 r + 49 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+19*r^2+49*r+45) The first few terms, starting at, 1, are [23, 94, 278, 673, 1421, 2716, 4812, 8031, 12771, 19514, 28834, 41405, 58009, 79544, 107032, 141627, 184623, 237462, 301742, 379225] ----------------------------------------------------------------- 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, r, r + 1, r + 3], is given by 3 2 (r + 2) (r + 1) (2 r + 19 r + 39 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+19*r^2+39*r+30) The first few terms, starting at, 1, are [18, 80, 248, 618, 1330, 2576, 4608, 7746, 12386, 19008, 28184, 40586, 56994, 78304, 105536, 139842, 182514, 234992, 298872, 375914] ----------------------------------------------------------------- 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, r, r, r + 4], 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] 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, r + 1, r + 1, r + 2], is given by 2 (2 r + 3) (r + 2) (r + 1) (r + 8 r + 10) ----------------------------------------- 30 and in Maple input form 1/30*(2*r+3)*(r+2)*(r+1)*(r^2+8*r+10) The first few terms, starting at, 1, are [19, 84, 258, 638, 1365, 2632, 4692, 7866, 12551, 19228, 28470, 40950, 57449, 78864, 106216, 140658, 183483, 236132, 300202, 377454] ----------------------------------------------------------------- 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, r, r + 2, r + 2], is given by 3 2 (r + 2) (r + 1) (2 r + 19 r + 39 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+19*r^2+39*r+30) The first few terms, starting at, 1, are [18, 80, 248, 618, 1330, 2576, 4608, 7746, 12386, 19008, 28184, 40586, 56994, 78304, 105536, 139842, 182514, 234992, 298872, 375914] ----------------------------------------------------------------- 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, r, r + 1, r + 3], is given by 4 3 2 (r + 1) (2 r + 23 r + 67 r + 103 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+23*r^3+67*r^2+103*r+60) The first few terms, starting at, 1, are [17, 75, 234, 588, 1275, 2485, 4468, 7542, 12101, 18623, 27678, 39936, 56175, 77289, 104296, 138346, 180729, 232883, 296402, 373044] ----------------------------------------------------------------- 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, r, r, r + 4], is given by 5 4 3 2 44 1/15 r + 5/6 r + 2 r + 25/6 r + -- r + 1 15 and in Maple input form 1/15*r^5+5/6*r^4+2*r^3+25/6*r^2+44/15*r+1 The first few terms, starting at, 1, are [11, 55, 185, 489, 1099, 2199, 4033, 6913, 11227, 17447, 26137, 37961, 53691, 74215, 100545, 133825, 175339, 226519, 288953, 364393] 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, r + 1, r + 1, r + 2], is given by 2 2 (r + 1) (r + 2 r + 2) (2 r + 19 r + 15) ----------------------------------------- 30 and in Maple input form 1/30*(r+1)*(r^2+2*r+2)*(2*r^2+19*r+15) The first few terms, starting at, 1, are [12, 61, 204, 533, 1184, 2345, 4264, 7257, 11716, 18117, 27028, 39117, 55160, 76049, 102800, 136561, 178620, 230413, 293532, 369733] ----------------------------------------------------------------- Proposition No. , 4.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r, r, r + 2, r + 2], 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. , 4.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 3], is given by 5 4 3 2 44 1/15 r + 5/6 r + 2 r + 25/6 r + -- r + 1 15 and in Maple input form 1/15*r^5+5/6*r^4+2*r^3+25/6*r^2+44/15*r+1 The first few terms, starting at, 1, are [11, 55, 185, 489, 1099, 2199, 4033, 6913, 11227, 17447, 26137, 37961, 53691, 74215, 100545, 133825, 175339, 226519, 288953, 364393] ----------------------------------------------------------------- Proposition No. , 4.24, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r] and Goals Against Vector, [r, r, r, r + 4], is given by 5 4 3 2 13 1/15 r + 5/6 r + r + 17/3 r + -- r + 2 30 and in Maple input form 1/15*r^5+5/6*r^4+r^3+17/3*r^2+13/30*r+2 The first few terms, starting at, 1, are [10, 49, 165, 440, 1000, 2023, 3747, 6478, 10598, 16573, 24961, 36420, 51716, 71731, 97471, 130074, 170818, 221129, 282589, 356944] This concludes this chapter, that took, 1.187, 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], [1, 1, 1, 2], 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, r + 1, r + 2, r + 2], is given by (4 r + 15) (r + 4) (r + 3) (r + 2) (r + 1) ------------------------------------------ 60 and in Maple input form 1/60*(4*r+15)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [38, 138, 378, 868, 1764, 3276, 5676, 9306, 14586, 22022, 32214, 45864, 63784, 86904, 116280, 153102, 198702, 254562, 322322, 403788] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 3], 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. , 5.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r, r, r + 2, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 20) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+20) The first few terms, starting at, 1, are [22, 98, 298, 728, 1540, 2940, 5196, 8646, 13706, 20878, 30758, 44044, 61544, 84184, 113016, 149226, 194142, 249242, 316162, 396704] ----------------------------------------------------------------- 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, r, r + 1, r + 4], 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. , 5.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r + 1] and Goals Against Vector, [r, r, r, r + 5], is given by 3 2 r (r + 1) (4 r + 51 r + 79 r + 46) ------------------------------------ 60 and in Maple input form 1/60*r*(r+1)*(4*r^3+51*r^2+79*r+46) The first few terms, starting at, 1, are [6, 44, 170, 478, 1108, 2254, 4172, 7188, 11706, 18216, 27302, 39650, 56056, 77434, 104824, 139400, 182478, 235524, 300162, 378182] 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 + 1, r + 1, r + 1, r + 2], is given by (4 r + 15) (r + 4) (r + 3) (r + 2) (r + 1) ------------------------------------------ 60 and in Maple input form 1/60*(4*r+15)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [38, 138, 378, 868, 1764, 3276, 5676, 9306, 14586, 22022, 32214, 45864, 63784, 86904, 116280, 153102, 198702, 254562, 322322, 403788] ----------------------------------------------------------------- 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, r + 1, r + 2, r + 2], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 50) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+50) The first few terms, starting at, 1, are [34, 128, 358, 833, 1708, 3192, 5556, 9141, 14366, 21736, 31850, 45409, 63224, 86224, 115464, 152133, 197562, 253232, 320782, 402017] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 3], 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. , 5.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r, r, r + 2, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+30) The first few terms, starting at, 1, are [26, 108, 318, 763, 1596, 3024, 5316, 8811, 13926, 21164, 31122, 44499, 62104, 84864, 113832, 150195, 195282, 250572, 317702, 398475] ----------------------------------------------------------------- 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, r, r + 1, r + 4], 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. , 5.11, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r] and Goals Against Vector, [r, r, r, r + 5], is given by 4 3 2 (r + 1) (4 r + 51 r + 109 r + 136 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+109*r^2+136*r+60) The first few terms, starting at, 1, are [12, 62, 210, 553, 1234, 2450, 4460, 7593, 12256, 18942, 28238, 40833, 57526, 79234, 107000, 142001, 185556, 239134, 304362, 383033] 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 + 1, r + 1, r + 1, r + 2], 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. , 5.13, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r, r + 1, r + 2, r + 2], 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. , 5.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r] and Goals Against Vector, [r, r + 1, r + 1, r + 3], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 123 r + 120) -------------------------------------------- 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+123*r+120) The first few terms, starting at, 1, are [29, 114, 328, 778, 1617, 3052, 5352, 8856, 13981, 21230, 31200, 44590, 62209, 84984, 113968, 150348, 195453, 250762, 317912, 398706] ----------------------------------------------------------------- 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, r, r + 2, r + 3], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 103 r + 60) ------------------------------------------- 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+103*r+60) The first few terms, starting at, 1, are [21, 94, 288, 708, 1505, 2884, 5112, 8526, 13541, 20658, 30472, 43680, 61089, 83624, 112336, 148410, 193173, 248102, 314832, 395164] ----------------------------------------------------------------- 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, r, r + 1, r + 4], is given by 4 3 2 (r + 1) (4 r + 51 r + 159 r + 236 r + 120) --------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+159*r^2+236*r+120) The first few terms, starting at, 1, are [19, 85, 264, 658, 1415, 2737, 4888, 8202, 13091, 20053, 29680, 42666, 59815, 82049, 110416, 146098, 190419, 244853, 311032, 390754] ----------------------------------------------------------------- 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, r, r, r + 5], is given by 5 11 4 3 55 2 31 1/15 r + -- r + 7/3 r + -- r + -- r + 1 12 12 10 and in Maple input form 1/15*r^5+11/12*r^4+7/3*r^3+55/12*r^2+31/10*r+1 The first few terms, starting at, 1, are [12, 61, 205, 539, 1204, 2395, 4369, 7453, 12052, 18657, 27853, 40327, 56876, 78415, 105985, 140761, 184060, 237349, 302253, 380563] 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 + 1, r + 1, r + 1, r + 2], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 20) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+20) The first few terms, starting at, 1, are [22, 98, 298, 728, 1540, 2940, 5196, 8646, 13706, 20878, 30758, 44044, 61544, 84184, 113016, 149226, 194142, 249242, 316162, 396704] ----------------------------------------------------------------- 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, r + 1, r + 2, r + 2], is given by 2 (r + 3) (r + 2) (r + 1) (4 r + 31 r + 30) ------------------------------------------ 60 and in Maple input form 1/60*(r+3)*(r+2)*(r+1)*(4*r^2+31*r+30) The first few terms, starting at, 1, are [26, 108, 318, 763, 1596, 3024, 5316, 8811, 13926, 21164, 31122, 44499, 62104, 84864, 113832, 150195, 195282, 250572, 317702, 398475] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 3], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 103 r + 60) ------------------------------------------- 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+103*r+60) The first few terms, starting at, 1, are [21, 94, 288, 708, 1505, 2884, 5112, 8526, 13541, 20658, 30472, 43680, 61089, 83624, 112336, 148410, 193173, 248102, 314832, 395164] ----------------------------------------------------------------- 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, r, r + 2, r + 3], is given by 3 2 (r + 2) (r + 1) (4 r + 43 r + 103 r + 90) ------------------------------------------- 60 and in Maple input form 1/60*(r+2)*(r+1)*(4*r^3+43*r^2+103*r+90) The first few terms, starting at, 1, are [24, 100, 298, 723, 1526, 2912, 5148, 8571, 13596, 20724, 30550, 43771, 61194, 83744, 112472, 148563, 193344, 248292, 315042, 395395] ----------------------------------------------------------------- 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, r, r + 1, r + 4], is given by 4 3 2 (r + 1) (4 r + 51 r + 149 r + 216 r + 120) --------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+149*r^2+216*r+120) The first few terms, starting at, 1, are [18, 81, 254, 638, 1380, 2681, 4804, 8082, 12926, 19833, 29394, 42302, 59360, 81489, 109736, 145282, 189450, 243713, 309702, 389214] ----------------------------------------------------------------- Proposition No. , 5.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r, r] and Goals Against Vector, [r, r, r, r + 5], is given by 5 11 4 3 49 2 44 1/15 r + -- r + 2 r + -- r + -- r + 1 12 12 15 and in Maple input form 1/15*r^5+11/12*r^4+2*r^3+49/12*r^2+44/15*r+1 The first few terms, starting at, 1, are [11, 56, 191, 509, 1149, 2304, 4229, 7249, 11767, 18272, 27347, 39677, 56057, 77400, 104745, 139265, 182275, 235240, 299783, 377693] If the Goals For Vector Was [r + 4, r + 1, r, r] ----------------------------------------------------------------- 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 + 1, r + 1, r + 1, r + 2], 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. , 5.25, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r, r + 1, r + 2, r + 2], 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. , 5.26, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 3], is given by 4 3 2 (r + 1) (4 r + 51 r + 159 r + 236 r + 120) --------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+159*r^2+236*r+120) The first few terms, starting at, 1, are [19, 85, 264, 658, 1415, 2737, 4888, 8202, 13091, 20053, 29680, 42666, 59815, 82049, 110416, 146098, 190419, 244853, 311032, 390754] ----------------------------------------------------------------- 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, r, r + 2, r + 3], is given by 4 3 2 (r + 1) (4 r + 51 r + 149 r + 216 r + 120) --------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+149*r^2+216*r+120) The first few terms, starting at, 1, are [18, 81, 254, 638, 1380, 2681, 4804, 8082, 12926, 19833, 29394, 42302, 59360, 81489, 109736, 145282, 189450, 243713, 309702, 389214] ----------------------------------------------------------------- Proposition No. , 5.28, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r, r, r + 1, r + 4], is given by 5 11 4 3 73 2 79 1/15 r + -- r + 8/3 r + -- r + -- r + 2 12 12 15 and in Maple input form 1/15*r^5+11/12*r^4+8/3*r^3+73/12*r^2+79/15*r+2 The first few terms, starting at, 1, are [17, 75, 235, 594, 1295, 2535, 4573, 7738, 12437, 19163, 28503, 41146, 57891, 79655, 107481, 142546, 186169, 239819, 305123, 383874] ----------------------------------------------------------------- Proposition No. , 5.29, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r, r] and Goals Against Vector, [r, r, r, r + 5], is given by 5 11 4 3 73 2 1/15 r + -- r + 4/3 r + -- r + 3/5 r + 2 12 12 and in Maple input form 1/15*r^5+11/12*r^4+4/3*r^3+73/12*r^2+3/5*r+2 The first few terms, starting at, 1, are [11, 55, 185, 490, 1105, 2219, 4083, 7018, 11423, 17783, 26677, 38786, 54901, 75931, 102911, 137010, 179539, 231959, 295889, 373114] If the Goals For Vector Was [r + 5, r, r, r] ----------------------------------------------------------------- Proposition No. , 5.30, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 2], is given by 3 2 r (r + 1) (4 r + 51 r + 79 r + 46) ------------------------------------ 60 and in Maple input form 1/60*r*(r+1)*(4*r^3+51*r^2+79*r+46) The first few terms, starting at, 1, are [6, 44, 170, 478, 1108, 2254, 4172, 7188, 11706, 18216, 27302, 39650, 56056, 77434, 104824, 139400, 182478, 235524, 300162, 378182] ----------------------------------------------------------------- Proposition No. , 5.31, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r, r + 1, r + 2, r + 2], is given by 4 3 2 (r + 1) (4 r + 51 r + 109 r + 136 r + 60) -------------------------------------------- 60 and in Maple input form 1/60*(r+1)*(4*r^4+51*r^3+109*r^2+136*r+60) The first few terms, starting at, 1, are [12, 62, 210, 553, 1234, 2450, 4460, 7593, 12256, 18942, 28238, 40833, 57526, 79234, 107000, 142001, 185556, 239134, 304362, 383033] ----------------------------------------------------------------- Proposition No. , 5.32, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 3], is given by 5 11 4 3 55 2 31 1/15 r + -- r + 7/3 r + -- r + -- r + 1 12 12 10 and in Maple input form 1/15*r^5+11/12*r^4+7/3*r^3+55/12*r^2+31/10*r+1 The first few terms, starting at, 1, are [12, 61, 205, 539, 1204, 2395, 4369, 7453, 12052, 18657, 27853, 40327, 56876, 78415, 105985, 140761, 184060, 237349, 302253, 380563] ----------------------------------------------------------------- Proposition No. , 5.33, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r, r, r + 2, r + 3], is given by 5 11 4 3 49 2 44 1/15 r + -- r + 2 r + -- r + -- r + 1 12 12 15 and in Maple input form 1/15*r^5+11/12*r^4+2*r^3+49/12*r^2+44/15*r+1 The first few terms, starting at, 1, are [11, 56, 191, 509, 1149, 2304, 4229, 7249, 11767, 18272, 27347, 39677, 56057, 77400, 104745, 139265, 182275, 235240, 299783, 377693] ----------------------------------------------------------------- Proposition No. , 5.34, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 4], is given by 5 11 4 3 73 2 1/15 r + -- r + 4/3 r + -- r + 3/5 r + 2 12 12 and in Maple input form 1/15*r^5+11/12*r^4+4/3*r^3+73/12*r^2+3/5*r+2 The first few terms, starting at, 1, are [11, 55, 185, 490, 1105, 2219, 4083, 7018, 11423, 17783, 26677, 38786, 54901, 75931, 102911, 137010, 179539, 231959, 295889, 373114] This concludes this chapter, that took, 2.356, 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], [1, 1, 2, 2], 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, r + 2, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) --------------------------------------- 15 and in Maple input form 1/15*(r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [48, 168, 448, 1008, 2016, 3696, 6336, 10296, 16016, 24024, 34944, 49504, 68544 , 93024, 124032, 162792, 210672, 269192, 340032, 425040] the smallest entry of, [2, 2, 1, 1], [1, 1, 1, 3], 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, r + 1, r + 2, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 18 r + 35) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+18*r+35) The first few terms, starting at, 1, are [44, 158, 428, 973, 1960, 3612, 6216, 10131, 15796, 23738, 34580, 49049, 67984, 92344, 123216, 161823, 209532, 267862, 338492, 423269] ----------------------------------------------------------------- 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, r, r + 3, r + 3], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 74 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+74*r+45) The first few terms, starting at, 1, are [29, 122, 358, 853, 1771, 3332, 5820, 9591, 15081, 22814, 33410, 47593, 66199, 90184, 120632, 158763, 205941, 263682, 333662, 417725] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 4], 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. , 6.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 5) -------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+5) The first few terms, starting at, 1, are [24, 108, 328, 798, 1680, 3192, 5616, 9306, 14696, 22308, 32760, 46774, 65184, 88944, 119136, 156978, 203832, 261212, 330792, 414414] ----------------------------------------------------------------- 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, r, r + 1, r + 5], 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. , 6.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 1, r + 1] and Goals Against Vector, [r, r, r, r + 6], is given by 3 2 r (r + 1) (r + 14 r + 16 r + 14) ---------------------------------- 15 and in Maple input form 1/15*r*(r+1)*(r^3+14*r^2+16*r+14) The first few terms, starting at, 1, are [6, 44, 172, 488, 1138, 2324, 4312, 7440, 12126, 18876, 28292, 41080, 58058, 80164, 108464, 144160, 188598, 243276, 309852, 390152] 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 + 1, r + 1, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) --------------------------------------- 15 and in Maple input form 1/15*(r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [48, 168, 448, 1008, 2016, 3696, 6336, 10296, 16016, 24024, 34944, 49504, 68544 , 93024, 124032, 162792, 210672, 269192, 340032, 425040] ----------------------------------------------------------------- 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, r + 2, r + 2, r + 2], 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. , 6.10, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 3], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) --------------------------------------- 15 and in Maple input form 1/15*(r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [48, 168, 448, 1008, 2016, 3696, 6336, 10296, 16016, 24024, 34944, 49504, 68544 , 93024, 124032, 162792, 210672, 269192, 340032, 425040] ----------------------------------------------------------------- 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, r + 1, r + 2, r + 3], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 84 r + 75) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+84*r+75) The first few terms, starting at, 1, are [37, 142, 398, 923, 1883, 3500, 6060, 9921, 15521, 23386, 34138, 48503, 67319, 91544, 122264, 160701, 208221, 266342, 336742, 421267] ----------------------------------------------------------------- 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, r, r + 3, r + 3], is given by 4 3 2 (r + 1) (r + 14 r + 56 r + 64 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+56*r^2+64*r+30) The first few terms, starting at, 1, are [22, 102, 316, 778, 1650, 3150, 5560, 9234, 14606, 22198, 32628, 46618, 65002, 88734, 118896, 156706, 203526, 260870, 330412, 413994] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 4], 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. , 6.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 4 3 2 (r + 1) (r + 14 r + 56 r + 79 r + 45) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+56*r^2+79*r+45) The first few terms, starting at, 1, are [26, 111, 332, 803, 1686, 3199, 5624, 9315, 14706, 22319, 32772, 46787, 65198, 88959, 119152, 156995, 203850, 261231, 330812, 414435] ----------------------------------------------------------------- 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, r, r + 1, r + 5], 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. , 6.16, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r + 2, r] and Goals Against Vector, [r, r, r, r + 6], is given by 4 3 2 (r + 1) (r + 14 r + 26 r + 34 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+26*r^2+34*r+15) The first few terms, starting at, 1, are [12, 63, 216, 573, 1284, 2555, 4656, 7929, 12796, 19767, 29448, 42549, 59892, 82419, 111200, 147441, 192492, 247855, 315192, 396333] If the Goals For Vector Was [r + 3, r + 1, r + 1, r + 1] the smallest entry of, [3, 1, 1, 1], [1, 1, 2, 2], 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, r + 2, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) --------------------------------------- 15 and in Maple input form 1/15*(r+5)*(r+4)*(r+3)*(r+2)*(r+1) The first few terms, starting at, 1, are [48, 168, 448, 1008, 2016, 3696, 6336, 10296, 16016, 24024, 34944, 49504, 68544 , 93024, 124032, 162792, 210672, 269192, 340032, 425040] the smallest entry of, [3, 1, 1, 1], [1, 1, 1, 3], 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, r + 1, r + 2, r + 3], 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. , 6.19, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r, r, r + 3, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 5) -------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+5) The first few terms, starting at, 1, are [24, 108, 328, 798, 1680, 3192, 5616, 9306, 14696, 22308, 32760, 46774, 65184, 88944, 119136, 156978, 203832, 261212, 330792, 414414] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 4], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 32 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+32*r+30) The first few terms, starting at, 1, are [30, 120, 348, 828, 1722, 3248, 5688, 9396, 14806, 22440, 32916, 46956, 65394, 89184, 119408, 157284, 204174, 261592, 331212, 414876] ----------------------------------------------------------------- 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, r, r + 2, r + 4], 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. , 6.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r + 1, r + 1] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 4 3 2 (r + 1) (2 r + 28 r + 72 r + 93 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+72*r^2+93*r+30) The first few terms, starting at, 1, are [15, 76, 250, 643, 1409, 2758, 4964, 8373, 13411, 20592, 30526, 43927, 61621, 84554, 113800, 150569, 196215, 252244, 320322, 402283] ----------------------------------------------------------------- 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, r, r, r + 6], is given by 4 3 2 r (r + 15 r + 25 r + 45 r + 4) --------------------------------- 15 and in Maple input form 1/15*r*(r^4+15*r^3+25*r^2+45*r+4) The first few terms, starting at, 1, are [6, 44, 170, 480, 1118, 2284, 4242, 7328, 11958, 18636, 27962, 40640, 57486, 79436, 107554, 143040, 187238, 241644, 307914, 387872] 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 + 1, r + 1, r + 2, r + 2], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 18 r + 35) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+18*r+35) The first few terms, starting at, 1, are [44, 158, 428, 973, 1960, 3612, 6216, 10131, 15796, 23738, 34580, 49049, 67984, 92344, 123216, 161823, 209532, 267862, 338492, 423269] ----------------------------------------------------------------- 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, r + 2, r + 2, r + 2], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 84 r + 75) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+84*r+75) The first few terms, starting at, 1, are [37, 142, 398, 923, 1883, 3500, 6060, 9921, 15521, 23386, 34138, 48503, 67319, 91544, 122264, 160701, 208221, 266342, 336742, 421267] ----------------------------------------------------------------- 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 + 1, r + 1, r + 1, r + 3], 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. , 6.27, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r, r + 1, r + 2, r + 3], 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. , 6.28, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r, r, r + 3, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 18 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+18*r+15) The first few terms, starting at, 1, are [28, 118, 348, 833, 1736, 3276, 5736, 9471, 14916, 22594, 33124, 47229, 65744, 89624, 119952, 157947, 204972, 262542, 332332, 416185] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 4], 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. , 6.30, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r, r, r + 2, r + 4], 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. , 6.31, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 4 3 2 (r + 1) (2 r + 28 r + 87 r + 123 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+87*r^2+123*r+60) The first few terms, starting at, 1, are [20, 91, 284, 708, 1520, 2933, 5224, 8742, 13916, 21263, 31396, 45032, 63000, 86249, 115856, 153034, 199140, 255683, 324332, 406924] ----------------------------------------------------------------- Proposition No. , 6.32, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 2, r + 1, r] and Goals Against Vector, [r, r, r, r + 6], is given by 5 4 3 2 31 1/15 r + r + 7/3 r + 9/2 r + -- r + 1 10 and in Maple input form 1/15*r^5+r^4+7/3*r^3+9/2*r^2+31/10*r+1 The first few terms, starting at, 1, are [12, 62, 211, 559, 1254, 2500, 4565, 7789, 12592, 19482, 29063, 42043, 59242, 81600, 110185, 146201, 190996, 246070, 313083, 393863] If the Goals For Vector Was [r + 3, r + 3, r, r] ----------------------------------------------------------------- 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 + 1, r + 1, r + 2, r + 2], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 74 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+74*r+45) The first few terms, starting at, 1, are [29, 122, 358, 853, 1771, 3332, 5820, 9591, 15081, 22814, 33410, 47593, 66199, 90184, 120632, 158763, 205941, 263682, 333662, 417725] ----------------------------------------------------------------- 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, r + 2, r + 2, r + 2], is given by 4 3 2 (r + 1) (r + 14 r + 56 r + 64 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+56*r^2+64*r+30) The first few terms, starting at, 1, are [22, 102, 316, 778, 1650, 3150, 5560, 9234, 14606, 22198, 32628, 46618, 65002, 88734, 118896, 156706, 203526, 260870, 330412, 413994] ----------------------------------------------------------------- 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 + 1, r + 1, r + 1, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 5) -------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+5) The first few terms, starting at, 1, are [24, 108, 328, 798, 1680, 3192, 5616, 9306, 14696, 22308, 32760, 46774, 65184, 88944, 119136, 156978, 203832, 261212, 330792, 414414] ----------------------------------------------------------------- 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, r + 1, r + 2, r + 3], is given by 2 (r + 3) (r + 2) (r + 1) (2 r + 18 r + 15) ------------------------------------------ 30 and in Maple input form 1/30*(r+3)*(r+2)*(r+1)*(2*r^2+18*r+15) The first few terms, starting at, 1, are [28, 118, 348, 833, 1736, 3276, 5736, 9471, 14916, 22594, 33124, 47229, 65744, 89624, 119952, 157947, 204972, 262542, 332332, 416185] ----------------------------------------------------------------- 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, r, r + 3, r + 3], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 32 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+32*r+30) The first few terms, starting at, 1, are [30, 120, 348, 828, 1722, 3248, 5688, 9396, 14806, 22440, 32916, 46956, 65394, 89184, 119408, 157284, 204174, 261592, 331212, 414876] ----------------------------------------------------------------- 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, r + 1, r + 1, r + 4], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 49 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+49*r+30) The first few terms, starting at, 1, are [21, 96, 298, 738, 1575, 3024, 5364, 8946, 14201, 21648, 31902, 45682, 63819, 87264, 117096, 154530, 200925, 257792, 326802, 409794] ----------------------------------------------------------------- Proposition No. , 6.39, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 54 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+54*r+45) The first few terms, starting at, 1, are [25, 106, 318, 773, 1631, 3108, 5484, 9111, 14421, 21934, 32266, 46137, 64379, 87944, 117912, 155499, 202065, 259122, 328342, 411565] ----------------------------------------------------------------- Proposition No. , 6.40, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 4 3 2 (r + 1) (r + 14 r + 36 r + 54 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+36*r^2+54*r+30) The first few terms, starting at, 1, are [18, 82, 260, 658, 1430, 2786, 5000, 8418, 13466, 20658, 30604, 44018, 61726, 84674, 113936, 150722, 196386, 252434, 320532, 402514] ----------------------------------------------------------------- Proposition No. , 6.41, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 3, r, r] and Goals Against Vector, [r, r, r, r + 6], 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] If the Goals For Vector Was [r + 4, r + 1, r + 1, r] ----------------------------------------------------------------- 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 + 1, r + 1, r + 2, r + 2], 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. , 6.43, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r + 2, r + 2, r + 2], 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. , 6.44, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 3], is given by 3 2 (r + 2) (r + 1) (r + 12 r + 32 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+2)*(r+1)*(r^3+12*r^2+32*r+30) The first few terms, starting at, 1, are [30, 120, 348, 828, 1722, 3248, 5688, 9396, 14806, 22440, 32916, 46956, 65394, 89184, 119408, 157284, 204174, 261592, 331212, 414876] ----------------------------------------------------------------- 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, r + 1, r + 2, r + 3], 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. , 6.46, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 3, r + 3], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 49 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+49*r+30) The first few terms, starting at, 1, are [21, 96, 298, 738, 1575, 3024, 5364, 8946, 14201, 21648, 31902, 45682, 63819, 87264, 117096, 154530, 200925, 257792, 326802, 409794] ----------------------------------------------------------------- Proposition No. , 6.47, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r + 1, r + 1, r + 4], is given by 4 3 2 (r + 1) (2 r + 28 r + 102 r + 183 r + 120) --------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+102*r^2+183*r+120) The first few terms, starting at, 1, are [29, 115, 334, 798, 1667, 3157, 5548, 9192, 14521, 22055, 32410, 46306, 64575, 88169, 118168, 155788, 202389, 259483, 328742, 412006] ----------------------------------------------------------------- Proposition No. , 6.48, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 4 3 2 (r + 1) (2 r + 28 r + 92 r + 133 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+92*r^2+133*r+60) The first few terms, starting at, 1, are [21, 95, 294, 728, 1555, 2989, 5308, 8862, 14081, 21483, 31682, 45396, 63455, 86809, 116536, 153850, 200109, 256823, 325662, 408464] ----------------------------------------------------------------- Proposition No. , 6.49, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 5 4 3 2 173 1/15 r + r + 19/6 r + 7 r + --- r + 2 30 and in Maple input form 1/15*r^5+r^4+19/6*r^3+7*r^2+173/30*r+2 The first few terms, starting at, 1, are [19, 85, 265, 664, 1435, 2787, 4993, 8398, 13427, 20593, 30505, 43876, 61531, 84415, 113601, 150298, 195859, 251789, 319753, 401584] ----------------------------------------------------------------- Proposition No. , 6.50, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 1, r + 1, r] and Goals Against Vector, [r, r, r, r + 6], is given by 5 4 3 2 23 1/15 r + r + 5/3 r + 13/2 r + -- r + 2 30 and in Maple input form 1/15*r^5+r^4+5/3*r^3+13/2*r^2+23/30*r+2 The first few terms, starting at, 1, are [12, 61, 205, 540, 1210, 2415, 4419, 7558, 12248, 18993, 28393, 41152, 58086, 80131, 108351, 143946, 188260, 242789, 309189, 389284] If the Goals For Vector Was [r + 4, r + 2, r, r] ----------------------------------------------------------------- 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 + 1, r + 1, r + 2, r + 2], is given by 2 (r + 3) (r + 2) (r + 1) (r + 9 r + 5) -------------------------------------- 15 and in Maple input form 1/15*(r+3)*(r+2)*(r+1)*(r^2+9*r+5) The first few terms, starting at, 1, are [24, 108, 328, 798, 1680, 3192, 5616, 9306, 14696, 22308, 32760, 46774, 65184, 88944, 119136, 156978, 203832, 261212, 330792, 414414] ----------------------------------------------------------------- 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, r + 2, r + 2, r + 2], is given by 4 3 2 (r + 1) (r + 14 r + 56 r + 79 r + 45) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+56*r^2+79*r+45) The first few terms, starting at, 1, are [26, 111, 332, 803, 1686, 3199, 5624, 9315, 14706, 22319, 32772, 46787, 65198, 88959, 119152, 156995, 203850, 261231, 330812, 414435] ----------------------------------------------------------------- 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 + 1, r + 1, r + 1, r + 3], 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. , 6.54, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r + 1, r + 2, r + 3], 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. , 6.55, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r, r + 3, r + 3], is given by 3 2 (r + 2) (r + 1) (2 r + 24 r + 54 r + 45) ------------------------------------------ 30 and in Maple input form 1/30*(r+2)*(r+1)*(2*r^3+24*r^2+54*r+45) The first few terms, starting at, 1, are [25, 106, 318, 773, 1631, 3108, 5484, 9111, 14421, 21934, 32266, 46137, 64379, 87944, 117912, 155499, 202065, 259122, 328342, 411565] ----------------------------------------------------------------- Proposition No. , 6.56, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 4], is given by 4 3 2 (r + 1) (2 r + 28 r + 92 r + 133 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+92*r^2+133*r+60) The first few terms, starting at, 1, are [21, 95, 294, 728, 1555, 2989, 5308, 8862, 14081, 21483, 31682, 45396, 63455, 86809, 116536, 153850, 200109, 256823, 325662, 408464] ----------------------------------------------------------------- Proposition No. , 6.57, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 4 3 2 (r + 1) (r + 14 r + 46 r + 74 r + 45) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+46*r^2+74*r+45) The first few terms, starting at, 1, are [24, 101, 304, 743, 1576, 3017, 5344, 8907, 14136, 21549, 31760, 45487, 63560, 86929, 116672, 154003, 200280, 257013, 325872, 408695] ----------------------------------------------------------------- Proposition No. , 6.58, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 5 4 3 2 163 1/15 r + r + 3 r + 13/2 r + --- r + 2 30 and in Maple input form 1/15*r^5+r^4+3*r^3+13/2*r^2+163/30*r+2 The first few terms, starting at, 1, are [18, 81, 255, 644, 1400, 2731, 4909, 8278, 13262, 20373, 30219, 43512, 61076, 83855, 112921, 149482, 194890, 250649, 318423, 400044] ----------------------------------------------------------------- Proposition No. , 6.59, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r + 2, r, r] and Goals Against Vector, [r, r, r, r + 6], is given by 5 4 3 2 1/15 r + r + 4/3 r + 6 r + 3/5 r + 2 and in Maple input form 1/15*r^5+r^4+4/3*r^3+6*r^2+3/5*r+2 The first few terms, starting at, 1, are [11, 56, 191, 510, 1155, 2324, 4279, 7354, 11963, 18608, 27887, 40502, 57267, 79116, 107111, 142450, 186475, 240680, 306719, 386414] If the Goals For Vector Was [r + 5, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 6.60, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 1, r + 1, r + 2, r + 2], 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. , 6.61, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r + 2, r + 2, r + 2], 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. , 6.62, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 3], is given by 4 3 2 (r + 1) (2 r + 28 r + 72 r + 93 r + 30) ------------------------------------------ 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+72*r^2+93*r+30) The first few terms, starting at, 1, are [15, 76, 250, 643, 1409, 2758, 4964, 8373, 13411, 20592, 30526, 43927, 61621, 84554, 113800, 150569, 196215, 252244, 320322, 402283] ----------------------------------------------------------------- Proposition No. , 6.63, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r + 1, r + 2, r + 3], is given by 4 3 2 (r + 1) (2 r + 28 r + 87 r + 123 r + 60) ------------------------------------------- 30 and in Maple input form 1/30*(r+1)*(2*r^4+28*r^3+87*r^2+123*r+60) The first few terms, starting at, 1, are [20, 91, 284, 708, 1520, 2933, 5224, 8742, 13916, 21263, 31396, 45032, 63000, 86249, 115856, 153034, 199140, 255683, 324332, 406924] ----------------------------------------------------------------- Proposition No. , 6.64, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r, r + 3, r + 3], is given by 4 3 2 (r + 1) (r + 14 r + 36 r + 54 r + 30) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+36*r^2+54*r+30) The first few terms, starting at, 1, are [18, 82, 260, 658, 1430, 2786, 5000, 8418, 13466, 20658, 30604, 44018, 61726, 84674, 113936, 150722, 196386, 252434, 320532, 402514] ----------------------------------------------------------------- Proposition No. , 6.65, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 4], is given by 5 4 3 2 173 1/15 r + r + 19/6 r + 7 r + --- r + 2 30 and in Maple input form 1/15*r^5+r^4+19/6*r^3+7*r^2+173/30*r+2 The first few terms, starting at, 1, are [19, 85, 265, 664, 1435, 2787, 4993, 8398, 13427, 20593, 30505, 43876, 61531, 84415, 113601, 150298, 195859, 251789, 319753, 401584] ----------------------------------------------------------------- Proposition No. , 6.66, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 5 4 3 2 163 1/15 r + r + 3 r + 13/2 r + --- r + 2 30 and in Maple input form 1/15*r^5+r^4+3*r^3+13/2*r^2+163/30*r+2 The first few terms, starting at, 1, are [18, 81, 255, 644, 1400, 2731, 4909, 8278, 13262, 20373, 30219, 43512, 61076, 83855, 112921, 149482, 194890, 250649, 318423, 400044] ----------------------------------------------------------------- Proposition No. , 6.67, : The number of possible Soccer Tournaments with Goals For vector, [r + 5, r + 1, r, r] and Goals Against Vector, [r, r, r + 1, r + 5], is given by 5 4 3 2 44 1/15 r + r + 2 r + 8 r + -- r + 3 15 and in Maple input form 1/15*r^5+r^4+2*r^3+8*r^2+44/15*r+3 The first few terms, starting at, 1, are [17, 75, 235, 595, 1301, 2555, 4623, 7843, 12633, 19499, 29043, 41971, 59101, 81371, 109847, 145731, 190369, 245259, 312059, 392595] If the Goals For Vector Was [r + 6, r, r, r] ----------------------------------------------------------------- Proposition No. , 6.68, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 1, r + 1, r + 2, r + 2], is given by 3 2 r (r + 1) (r + 14 r + 16 r + 14) ---------------------------------- 15 and in Maple input form 1/15*r*(r+1)*(r^3+14*r^2+16*r+14) The first few terms, starting at, 1, are [6, 44, 172, 488, 1138, 2324, 4312, 7440, 12126, 18876, 28292, 41080, 58058, 80164, 108464, 144160, 188598, 243276, 309852, 390152] ----------------------------------------------------------------- Proposition No. , 6.69, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r + 2, r + 2, r + 2], is given by 4 3 2 (r + 1) (r + 14 r + 26 r + 34 r + 15) ---------------------------------------- 15 and in Maple input form 1/15*(r+1)*(r^4+14*r^3+26*r^2+34*r+15) The first few terms, starting at, 1, are [12, 63, 216, 573, 1284, 2555, 4656, 7929, 12796, 19767, 29448, 42549, 59892, 82419, 111200, 147441, 192492, 247855, 315192, 396333] ----------------------------------------------------------------- Proposition No. , 6.70, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r + 1, r + 1, r + 1, r + 3], is given by 4 3 2 r (r + 15 r + 25 r + 45 r + 4) --------------------------------- 15 and in Maple input form 1/15*r*(r^4+15*r^3+25*r^2+45*r+4) The first few terms, starting at, 1, are [6, 44, 170, 480, 1118, 2284, 4242, 7328, 11958, 18636, 27962, 40640, 57486, 79436, 107554, 143040, 187238, 241644, 307914, 387872] ----------------------------------------------------------------- Proposition No. , 6.71, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r + 1, r + 2, r + 3], is given by 5 4 3 2 31 1/15 r + r + 7/3 r + 9/2 r + -- r + 1 10 and in Maple input form 1/15*r^5+r^4+7/3*r^3+9/2*r^2+31/10*r+1 The first few terms, starting at, 1, are [12, 62, 211, 559, 1254, 2500, 4565, 7789, 12592, 19482, 29063, 42043, 59242, 81600, 110185, 146201, 190996, 246070, 313083, 393863] ----------------------------------------------------------------- Proposition No. , 6.72, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r, r + 3, r + 3], 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. , 6.73, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 4], is given by 5 4 3 2 23 1/15 r + r + 5/3 r + 13/2 r + -- r + 2 30 and in Maple input form 1/15*r^5+r^4+5/3*r^3+13/2*r^2+23/30*r+2 The first few terms, starting at, 1, are [12, 61, 205, 540, 1210, 2415, 4419, 7558, 12248, 18993, 28393, 41152, 58086, 80131, 108351, 143946, 188260, 242789, 309189, 389284] ----------------------------------------------------------------- Proposition No. , 6.74, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r, r + 2, r + 4], is given by 5 4 3 2 1/15 r + r + 4/3 r + 6 r + 3/5 r + 2 and in Maple input form 1/15*r^5+r^4+4/3*r^3+6*r^2+3/5*r+2 The first few terms, starting at, 1, are [11, 56, 191, 510, 1155, 2324, 4279, 7354, 11963, 18608, 27887, 40502, 57267, 79116, 107111, 142450, 186475, 240680, 306719, 386414] ----------------------------------------------------------------- Proposition No. , 6.75, : The number of possible Soccer Tournaments with Goals For vector, [r + 6, r, r, r] and Goals Against Vector, [r, r, r, r + 6], is given by 5 4 3 2 r (r - 81 r - 275 r - 2175 r - 806 r - 3864) - ------------------------------------------------ 720 and in Maple input form -1/720*r*(r^5-81*r^4-275*r^3-2175*r^2-806*r-3864) The first few terms, starting at, 1, are [10, 49, 165, 440, 1001, 2030, 3773, 6548, 10752, 16867, 25465, 37212, 52871, 73304, 99473, 132440, 173366, 223509, 284221, 356944] This concludes this chapter, that took, 4.747, seconds to produce. 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