Polynomial Expressions for the Number of Round-Robin Tournaments with , 5, Teams, whose excess is <=, 4 By Shalosh B. Ekhad ---------------------------------------------------- Chapter Number, 0 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 5, 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, r] ----------------------------------------------------------------- Proposition No. , 0.1, : The number of possible Soccer Tournaments with Goals For vector, [r, r, r, r, r] and Goals Against Vector, [r, r, r, r, r], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (43 r + 688 r + 4934 r + 20680 r + 55907 r 3 2 + 101272 r + 123436 r + 96240 r + 40320)/241920 and in Maple input form 1/241920*(r+3)*(r+2)*(r+1)*(43*r^8+688*r^7+4934*r^6+20680*r^5+55907*r^4+101272* r^3+123436*r^2+96240*r+40320) The first few terms, starting at, 1, are [44, 870, 9480, 68290, 365936, 1573374, 5709120, 18107760, 51488800, 133748186, 321979164, 726436425, 1549758640, 3148837580, 6129352176, 11486265339, 20807609460, 36563769670, 62510325680, 104239453956] This concludes this chapter, that took, 110.961, seconds to produce. ---------------------------------------------------- Chapter Number, 1 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 5, 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, r] ----------------------------------------------------------------- Proposition No. , 1.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r, r, r, r] and Goals Against Vector, [r, r, r, r, r + 1], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 3053 r + 18855 r + 65425 r 3 2 + 138430 r + 180522 r + 137820 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+3053*r^6+18855*r^5+65425*r^4+138430* r^3+180522*r^2+137820*r+50400) The first few terms, starting at, 1, are [59, 1199, 12780, 89491, 467172, 1963797, 6990027, 21809634, 61147801, 156915902, 373763039, 835442634, 1767669112, 3565314630, 6894545466, 12843915735, 23142623295, 40469196889, 68879939678, 114393624389] This concludes this chapter, that took, 137.348, seconds to produce. ---------------------------------------------------- Chapter Number, 2 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 5, 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, r] ----------------------------------------------------------------- Proposition No. , 2.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 1], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 3526 r + 24810 r + 97140 r 3 2 + 229015 r + 327014 r + 265800 r + 100800)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+3526*r^6+24810*r^5+97140*r^4+229015* r^3+327014*r^2+265800*r+100800) The first few terms, starting at, 1, are [104, 1897, 18724, 124016, 620916, 2526861, 8764932, 26777916, 73776560, 186552938, 438792796, 970173880, 2033314376, 4066917582, 7806333768, 14446322934, 25875155016, 45004436279, 76225318532, 126029154856] ----------------------------------------------------------------- Proposition No. , 2.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 1) 5 4 3 2 2 (215 r + 2021 r + 7563 r + 13839 r + 12762 r + 5040) (r + 2) /1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^5+2021*r^4+7563*r^3+13839*r^2+12762*r+ 5040)*(r+2)^2 The first few terms, starting at, 1, are [74, 1547, 16334, 112431, 576660, 2384976, 8367282, 25776036, 71461390, 181572677, 428706356, 950770730, 1997611656, 4003721952, 7698228252, 14266907739, 25585335666, 44547512119, 75520579442, 124963643585] If the Goals For Vector Was [r + 2, r, r, r, r] ----------------------------------------------------------------- Proposition No. , 2.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 1], is given by (r + 5) (r + 4) (r + 3) (r + 1) 5 4 3 2 2 (215 r + 2021 r + 7563 r + 13839 r + 12762 r + 5040) (r + 2) /1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^5+2021*r^4+7563*r^3+13839*r^2+12762*r+ 5040)*(r+2)^2 The first few terms, starting at, 1, are [74, 1547, 16334, 112431, 576660, 2384976, 8367282, 25776036, 71461390, 181572677, 428706356, 950770730, 1997611656, 4003721952, 7698228252, 14266907739, 25585335666, 44547512119, 75520579442, 124963643585] ----------------------------------------------------------------- Proposition No. , 2.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r, r, r, r] and Goals Against Vector, [r, r, r, r, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2451 r + 10655 r + 23565 r + 29930 r + 23904 r + 10080)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2451*r^5+10655*r^4+23565*r^3+ 29930*r^2+23904*r+10080) The first few terms, starting at, 1, are [60, 1310, 14460, 102636, 537540, 2255913, 7998408, 24833523, 69260620, 176800767, 418981784, 931971144, 1962881064, 3942045102, 7592432832, 14090923773, 25300505868, 44097706708, 74825823732, 123911916260] This concludes this chapter, that took, 295.297, seconds to produce. ---------------------------------------------------- Chapter Number, 3 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 5, 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, r] ----------------------------------------------------------------- Proposition No. , 3.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 1], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 3999 r + 31905 r + 141485 r 3 2 + 376840 r + 602836 r + 536160 r + 201600)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+3999*r^6+31905*r^5+141485*r^4+376840 *r^3+602836*r^2+536160*r+201600) The first few terms, starting at, 1, are [188, 3056, 27784, 173476, 831177, 3269748, 11040744, 33001980, 89295349, 222385592, 516334130, 1128913864, 2343042238, 4646403144, 8851132824, 16269157284, 28963292466, 50099664208, 84433490636, 138967919948] ----------------------------------------------------------------- Proposition No. , 3.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2924 r + 16645 r + 50320 r + 85200 r + 76536 r + 30240)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2924*r^5+16645*r^4+50320*r^3+ 85200*r^2+76536*r+30240) The first few terms, starting at, 1, are [156, 2709, 25560, 163200, 793323, 3151764, 10717410, 32202027, 87474387, 218517585, 508584024, 1114142328, 2316079626, 4599013536, 8770570164, 16136194011, 28749576438, 49764231957, 83918233872, 138191767560] ----------------------------------------------------------------- Proposition No. , 3.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r, r, r + 3], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 4859 r + 45361 r + 228155 r + 678820 r 3 2 + 1225106 r + 1319324 r + 782160 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+4859*r^7+45361*r^6+228155*r^5+678820*r^4+ 1225106*r^3+1319324*r^2+782160*r+201600) The first few terms, starting at, 1, are [89, 1914, 20142, 137110, 694400, 2836911, 9840885, 30006966, 82429567, 207718511, 486809115, 1072420713, 2239586272, 4464059546, 8540400534, 15755241351, 28135746573, 48798715360, 82432244972, 135949511544] If the Goals For Vector Was [r + 2, r + 1, r, r, r] ----------------------------------------------------------------- Proposition No. , 3.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 1], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2924 r + 16645 r + 50320 r + 85200 r + 76536 r + 30240)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2924*r^5+16645*r^4+50320*r^3+ 85200*r^2+76536*r+30240) The first few terms, starting at, 1, are [156, 2709, 25560, 163200, 793323, 3151764, 10717410, 32202027, 87474387, 218517585, 508584024, 1114142328, 2316079626, 4599013536, 8770570164, 16136194011, 28749576438, 49764231957, 83918233872, 138191767560] ----------------------------------------------------------------- Proposition No. , 3.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2924 r + 15695 r + 43060 r + 64670 r + 53196 r + 20160)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2924*r^5+15695*r^4+43060*r^3+ 64670*r^2+53196*r+20160) The first few terms, starting at, 1, are [119, 2267, 22572, 148953, 739791, 2982705, 10249866, 31037721, 84811298, 212840342, 497177408, 1092354822, 2276243322, 4528902510, 8651251380, 15939092628, 28432540785, 49266342125, 83153051564, 137038673255] ----------------------------------------------------------------- Proposition No. , 3.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r, r + 3], is given by (r + 4) (r + 3) (r + 1) 6 5 4 3 2 (215 r + 3569 r + 21277 r + 56131 r + 79428 r + 66180 r + 25200) 2 (r + 2) /1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+1)*(215*r^6+3569*r^5+21277*r^4+56131*r^3+79428*r^2+ 66180*r+25200)*(r+2)^2 The first few terms, starting at, 1, are [75, 1658, 18061, 126128, 650454, 2692152, 9428463, 28957269, 79988623, 202447674, 476110856, 1051818040, 2201662724, 4396942560, 8425642986, 15564923937, 27828587139, 48314930026, 81686859639, 134823770840] If the Goals For Vector Was [r + 3, r, r, r, r] ----------------------------------------------------------------- Proposition No. , 3.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 1], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 4859 r + 45361 r + 228155 r + 678820 r 3 2 + 1225106 r + 1319324 r + 782160 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+4859*r^7+45361*r^6+228155*r^5+678820*r^4+ 1225106*r^3+1319324*r^2+782160*r+201600) The first few terms, starting at, 1, are [89, 1914, 20142, 137110, 694400, 2836911, 9840885, 30006966, 82429567, 207718511, 486809115, 1072420713, 2239586272, 4464059546, 8540400534, 15755241351, 28135746573, 48798715360, 82432244972, 135949511544] ----------------------------------------------------------------- Proposition No. , 3.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 2], is given by (r + 4) (r + 3) (r + 1) 6 5 4 3 2 (215 r + 3569 r + 21277 r + 56131 r + 79428 r + 66180 r + 25200) 2 (r + 2) /1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+1)*(215*r^6+3569*r^5+21277*r^4+56131*r^3+79428*r^2+ 66180*r+25200)*(r+2)^2 The first few terms, starting at, 1, are [75, 1658, 18061, 126128, 650454, 2692152, 9428463, 28957269, 79988623, 202447674, 476110856, 1051818040, 2201662724, 4396942560, 8425642986, 15564923937, 27828587139, 48314930026, 81686859639, 134823770840] ----------------------------------------------------------------- Proposition No. , 3.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r, r, r, r] and Goals Against Vector, [r, r, r, r, r + 3], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 4859 r + 42511 r + 181895 r + 431710 r 3 2 + 723806 r + 859724 r + 577680 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+4859*r^7+42511*r^6+181895*r^5+431710*r^4+ 723806*r^3+859724*r^2+577680*r+201600) The first few terms, starting at, 1, are [60, 1325, 15021, 108888, 578091, 2445663, 8708742, 27090954, 75585103, 192827778, 456399593, 1013559651, 2130775674, 4270784370, 8208899412, 15203973375, 27243918702, 47391111931, 80259462789, 132662550218] This concludes this chapter, that took, 565.190, seconds to produce. ---------------------------------------------------- Chapter Number, 4 Polynomial Expressions for the Number of Ways a Soccer Tournmaent Round-Robi\ n tournament with, 5, 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, r] ----------------------------------------------------------------- Proposition No. , 4.1, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r + 1, r + 1, r + 1, r + 1], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 40140 r + 201700 r 3 2 + 614305 r + 1138428 r + 1196100 r + 554400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+40140*r^5+201700*r^4+614305 *r^3+1138428*r^2+1196100*r+554400) The first few terms, starting at, 1, are [372, 5131, 42220, 246455, 1124889, 4265700, 13995663, 40878486, 108525131, 266013033, 609346309, 1316903146, 2705786198, 5318500728, 10052524218, 18349198149, 32463072630, 55838479631, 93626819946, 153385948969] ----------------------------------------------------------------- Proposition No. , 4.2, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 22515 r + 79715 r + 158350 r + 165888 r + 70560)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+22515*r^4+79715*r^3+ 158350*r^2+165888*r+70560) The first few terms, starting at, 1, are [298, 4492, 38620, 231156, 1071750, 4107147, 13575540, 39866541, 106271308, 261311479, 600068872, 1299449944, 2674286388, 5263679706, 9960133944, 18197884923, 32221531254, 55461714286, 93051300100, 152523413020] ----------------------------------------------------------------- Proposition No. , 4.3, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r, r, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 1) 5 4 3 2 2 (215 r + 2967 r + 15631 r + 38913 r + 43074 r + 15120) (r + 2) /1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^5+2967*r^4+15631*r^3+38913*r^2+43074*r +15120)*(r+2)^2 The first few terms, starting at, 1, are [207, 3636, 33588, 209259, 994602, 3874860, 12956328, 38368902, 102926109, 254318922, 586250184, 1273425426, 2627279892, 5181824400, 9822125088, 17971792029, 31860544155, 54898555008, 92190984732, 151233994725] ----------------------------------------------------------------- Proposition No. , 4.4, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r, r, r + 1, r + 3], is given by (r + 6) (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2107 r + 8283 r + 15857 r + 14578 r + 5040)/1209600 and in Maple input form 1/1209600*(r+6)*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2107*r^4+8283*r^3+15857* r^2+14578*r+5040) The first few terms, starting at, 1, are [192, 3408, 31902, 200841, 962082, 3770250, 12663222, 37631880, 101227698, 250676712, 578897124, 1259324742, 2601412284, 5136168996, 9744236868, 17842856889, 31652768268, 54571717464, 91687951410, 150474956775] ----------------------------------------------------------------- Proposition No. , 4.5, : The number of possible Soccer Tournaments with Goals For vector, [r + 1, r + 1, r + 1, r + 1, r] and Goals Against Vector, [r, r, r, r, r + 4], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 53258 r + 278020 r + 836915 r 3 2 + 1488088 r + 1536012 r + 842160 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+53258*r^6+278020*r^5+836915*r^4+ 1488088*r^3+1536012*r^2+842160*r+201600) The first few terms, starting at, 1, are [104, 2300, 24204, 163528, 820392, 3319602, 11410836, 34502424, 94052332, 235353404, 548071316, 1200392583, 2493592960, 4946327412, 9421062312, 17308916571, 30793856016, 53222806612, 89614936268, 147351228266] If the Goals For Vector Was [r + 2, r + 1, r + 1, r, r] ----------------------------------------------------------------- Proposition No. , 4.6, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 1, r + 1], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 22515 r + 79715 r + 158350 r + 165888 r + 70560)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+22515*r^4+79715*r^3+ 158350*r^2+165888*r+70560) The first few terms, starting at, 1, are [298, 4492, 38620, 231156, 1071750, 4107147, 13575540, 39866541, 106271308, 261311479, 600068872, 1299449944, 2674286388, 5263679706, 9960133944, 18197884923, 32221531254, 55461714286, 93051300100, 152523413020] ----------------------------------------------------------------- Proposition No. , 4.7, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 1) 5 4 3 2 2 (215 r + 2967 r + 15941 r + 41623 r + 52374 r + 25200) (r + 2) /1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^5+2967*r^4+15941*r^3+41623*r^2+52374*r +25200)*(r+2)^2 The first few terms, starting at, 1, are [247, 3986, 35553, 217545, 1023141, 3959340, 13178583, 38900862, 104104286, 256764508, 591054854, 1282428966, 2643472902, 5209917648, 9869336706, 18048914739, 31983367185, 55089729398, 92482436431, 151670007335] ----------------------------------------------------------------- Proposition No. , 4.8, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 21235 r + 67295 r + 114570 r + 102408 r + 40320)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+21235*r^4+67295*r^3+ 114570*r^2+102408*r+40320) The first few terms, starting at, 1, are [208, 3550, 32766, 204816, 976884, 3817077, 12793506, 37958778, 101981308, 252295615, 582173332, 1265624568, 2613002136, 5156682726, 9779330172, 17901104448, 31746870936, 54720099940, 91916845570, 150821082280] ----------------------------------------------------------------- Proposition No. , 4.9, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 3], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 37270 r + 160770 r 3 2 + 391875 r + 549358 r + 428520 r + 151200)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+37270*r^5+160770*r^4+391875 *r^3+549358*r^2+428520*r+151200) The first few terms, starting at, 1, are [171, 3101, 29702, 190134, 921585, 3642273, 12310023, 36755235, 99230274, 246435618, 570409931, 1243176116, 2571994974, 5084576070, 9656722602, 17698741395, 31421635605, 54209728419, 91133060172, 149640785162] ----------------------------------------------------------------- Proposition No. , 4.10, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 1, r + 1, r, r] and Goals Against Vector, [r, r, r, r, r + 4], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 52308 r + 260350 r + 730245 r 3 2 + 1230718 r + 1282912 r + 772320 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+52308*r^6+260350*r^5+730245*r^4+ 1230718*r^3+1282912*r^2+772320*r+201600) The first few terms, starting at, 1, are [90, 2025, 21916, 151359, 771620, 3159147, 10954866, 33345543, 91371214, 229583640, 536399370, 1177986823, 2452476456, 4873770290, 9297342636, 17104265709, 30464366946, 52705041355, 88818921312, 146151474183] If the Goals For Vector Was [r + 2, r + 2, r, r, r] ----------------------------------------------------------------- Proposition No. , 4.11, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 1, r + 1], is given by (r + 5) (r + 4) (r + 3) (r + 1) 5 4 3 2 2 (215 r + 2967 r + 15631 r + 38913 r + 43074 r + 15120) (r + 2) /1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^5+2967*r^4+15631*r^3+38913*r^2+43074*r +15120)*(r+2)^2 The first few terms, starting at, 1, are [207, 3636, 33588, 209259, 994602, 3874860, 12956328, 38368902, 102926109, 254318922, 586250184, 1273425426, 2627279892, 5181824400, 9822125088, 17971792029, 31860544155, 54898555008, 92190984732, 151233994725] ----------------------------------------------------------------- Proposition No. , 4.12, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 21235 r + 67295 r + 114570 r + 102408 r + 40320)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+21235*r^4+67295*r^3+ 114570*r^2+102408*r+40320) The first few terms, starting at, 1, are [208, 3550, 32766, 204816, 976884, 3817077, 12793506, 37958778, 101981308, 252295615, 582173332, 1265624568, 2613002136, 5156682726, 9779330172, 17901104448, 31746870936, 54720099940, 91916845570, 150821082280] ----------------------------------------------------------------- Proposition No. , 4.13, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r, r] and Goals Against Vector, [r, r, r, r + 2, r + 2], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 20285 r + 58175 r + 88340 r + 74868 r + 30240)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+20285*r^4+58175*r^3+ 88340*r^2+74868*r+30240) The first few terms, starting at, 1, are [164, 3012, 29144, 187731, 913452, 3619044, 12251580, 36622014, 98949708, 245881922, 569375040, 1241329154, 2568827160, 5079327456, 9648286488, 17685541677, 31401471684, 54179581632, 91088855000, 149577097725] ----------------------------------------------------------------- Proposition No. , 4.14, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 3], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36320 r + 146900 r 3 2 + 326285 r + 425708 r + 310020 r + 100800)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36320*r^5+146900*r^4+326285 *r^3+425708*r^2+310020*r+100800) The first few terms, starting at, 1, are [134, 2637, 26477, 174545, 862659, 3455844, 11794695, 35474175, 96307211, 240221267, 557960130, 1219464974, 2528764994, 5008702248, 9527938626, 17486542992, 31081138524, 53676216209, 90314909531, 148410403911] ----------------------------------------------------------------- Proposition No. , 4.15, : The number of possible Soccer Tournaments with Goals For vector, [r + 2, r + 2, r, r, r] and Goals Against Vector, [r, r, r, r, r + 4], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 51358 r + 242680 r + 627895 r 3 2 + 1024708 r + 1037012 r + 639600 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+51358*r^6+242680*r^5+627895*r^4+ 1024708*r^3+1037012*r^2+639600*r+201600) The first few terms, starting at, 1, are [76, 1769, 19788, 139929, 725344, 3005598, 10515504, 32224632, 88761904, 223948153, 524965376, 1155984180, 2412017232, 4802250100, 9175212528, 16901990835, 30138350772, 52192251913, 88029908692, 144961415819] If the Goals For Vector Was [r + 3, r + 1, r, r, r] ----------------------------------------------------------------- Proposition No. , 4.16, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 1, r + 1], is given by (r + 6) (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2107 r + 8283 r + 15857 r + 14578 r + 5040)/1209600 and in Maple input form 1/1209600*(r+6)*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2107*r^4+8283*r^3+15857* r^2+14578*r+5040) The first few terms, starting at, 1, are [192, 3408, 31902, 200841, 962082, 3770250, 12663222, 37631880, 101227698, 250676712, 578897124, 1259324742, 2601412284, 5136168996, 9744236868, 17842856889, 31652768268, 54571717464, 91687951410, 150474956775] ----------------------------------------------------------------- Proposition No. , 4.17, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 2], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 37270 r + 160770 r 3 2 + 391875 r + 549358 r + 428520 r + 151200)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+37270*r^5+160770*r^4+391875 *r^3+549358*r^2+428520*r+151200) The first few terms, starting at, 1, are [171, 3101, 29702, 190134, 921585, 3642273, 12310023, 36755235, 99230274, 246435618, 570409931, 1243176116, 2571994974, 5084576070, 9656722602, 17698741395, 31421635605, 54209728419, 91133060172, 149640785162] ----------------------------------------------------------------- Proposition No. , 4.18, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r + 2, r + 2], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36320 r + 146900 r 3 2 + 326285 r + 425708 r + 310020 r + 100800)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36320*r^5+146900*r^4+326285 *r^3+425708*r^2+310020*r+100800) The first few terms, starting at, 1, are [134, 2637, 26477, 174545, 862659, 3455844, 11794695, 35474175, 96307211, 240221267, 557960130, 1219464974, 2528764994, 5008702248, 9527938626, 17486542992, 31081138524, 53676216209, 90314909531, 148410403911] ----------------------------------------------------------------- Proposition No. , 4.19, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 3], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 53258 r + 275680 r + 816395 r 3 2 + 1498348 r + 1764612 r + 1230960 r + 403200)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+53258*r^6+275680*r^5+816395*r^4+ 1498348*r^3+1764612*r^2+1230960*r+403200) The first few terms, starting at, 1, are [120, 2378, 24350, 163254, 817338, 3306348, 11368602, 34389861, 93787034, 234783120, 546931112, 1198243397, 2489736108, 4939688640, 9410037948, 17291173212, 30766073988, 53180356166, 89551474122, 147258200672] ----------------------------------------------------------------- Proposition No. , 4.20, : The number of possible Soccer Tournaments with Goals For vector, [r + 3, r + 1, r, r, r] and Goals Against Vector, [r, r, r, r, r + 4], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 50408 r + 226180 r + 549125 r 3 2 + 978928 r + 1085412 r + 682800 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+50408*r^6+226180*r^5+549125*r^4+ 978928*r^3+1085412*r^2+682800*r+201600) The first few terms, starting at, 1, are [75, 1673, 18622, 132475, 692029, 2887848, 10163577, 31297860, 86552026, 219084723, 514947797, 1136467462, 2375757814, 4737593728, 9063977958, 16716570747, 29837814453, 51717192709, 87295732656, 143849716109] If the Goals For Vector Was [r + 4, r, r, r, r] ----------------------------------------------------------------- Proposition No. , 4.21, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r, r] and Goals Against Vector, [r, r + 1, r + 1, r + 1, r + 1], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 53258 r + 278020 r + 836915 r 3 2 + 1488088 r + 1536012 r + 842160 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+53258*r^6+278020*r^5+836915*r^4+ 1488088*r^3+1536012*r^2+842160*r+201600) The first few terms, starting at, 1, are [104, 2300, 24204, 163528, 820392, 3319602, 11410836, 34502424, 94052332, 235353404, 548071316, 1200392583, 2493592960, 4946327412, 9421062312, 17308916571, 30793856016, 53222806612, 89614936268, 147351228266] ----------------------------------------------------------------- Proposition No. , 4.22, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r, r] and Goals Against Vector, [r, r, r + 1, r + 1, r + 2], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 52308 r + 260350 r + 730245 r 3 2 + 1230718 r + 1282912 r + 772320 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+52308*r^6+260350*r^5+730245*r^4+ 1230718*r^3+1282912*r^2+772320*r+201600) The first few terms, starting at, 1, are [90, 2025, 21916, 151359, 771620, 3159147, 10954866, 33345543, 91371214, 229583640, 536399370, 1177986823, 2452476456, 4873770290, 9297342636, 17104265709, 30464366946, 52705041355, 88818921312, 146151474183] ----------------------------------------------------------------- Proposition No. , 4.23, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r, r] and Goals Against Vector, [r, r, r, r + 2, r + 2], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 51358 r + 242680 r + 627895 r 3 2 + 1024708 r + 1037012 r + 639600 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+51358*r^6+242680*r^5+627895*r^4+ 1024708*r^3+1037012*r^2+639600*r+201600) The first few terms, starting at, 1, are [76, 1769, 19788, 139929, 725344, 3005598, 10515504, 32224632, 88761904, 223948153, 524965376, 1155984180, 2412017232, 4802250100, 9175212528, 16901990835, 30138350772, 52192251913, 88029908692, 144961415819] ----------------------------------------------------------------- Proposition No. , 4.24, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r, r] and Goals Against Vector, [r, r, r, r + 1, r + 3], is given by 8 7 6 5 4 (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 50408 r + 226180 r + 549125 r 3 2 + 978928 r + 1085412 r + 682800 r + 201600)/1209600 and in Maple input form 1/1209600*(r+3)*(r+2)*(r+1)*(215*r^8+5332*r^7+50408*r^6+226180*r^5+549125*r^4+ 978928*r^3+1085412*r^2+682800*r+201600) The first few terms, starting at, 1, are [75, 1673, 18622, 132475, 692029, 2887848, 10163577, 31297860, 86552026, 219084723, 514947797, 1136467462, 2375757814, 4737593728, 9063977958, 16716570747, 29837814453, 51717192709, 87295732656, 143849716109] ----------------------------------------------------------------- Proposition No. , 4.25, : The number of possible Soccer Tournaments with Goals For vector, [r + 4, r, r, r, r] and Goals Against Vector, [r, r, r, r, r + 4], is given by 10 9 8 7 6 5 (r + 1) (215 r + 6407 r + 75508 r + 446462 r + 1506763 r + 3746303 r 4 3 2 + 7669922 r + 9274908 r + 7759752 r + 4592160 r + 1209600)/1209600 and in Maple input form 1/1209600*(r+1)*(215*r^10+6407*r^9+75508*r^8+446462*r^7+1506763*r^6+3746303*r^5 +7669922*r^4+9274908*r^3+7759752*r^2+4592160*r+1209600) The first few terms, starting at, 1, are [60, 1326, 15132, 111099, 597272, 2552406, 9156684, 28633536, 80170084, 204982404, 485796708, 1079498238, 2269632756, 4547914730, 8736992472, 16170541491, 28951408188, 50314118842, 85124722812, 140558780301] This concludes this chapter, that took, 1177.789, seconds to produce. This ends this wonderful webbook that took, 2286.585, seconds.