All the Polynomial Expressions for the Number of Round-Robin Tournaments who\ se 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, 111.185, 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 + 1, r, r, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3053 r + 18545 r + 62135 r + 122960 r + 140372 r + 76080)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3053*r^5+18545*r^4+62135*r^3+ 122960*r^2+140372*r+76080) The first few terms, starting at, 1, are [42, 1044, 11838, 85200, 451368, 1914288, 6853407, 21469140, 60367164, 155246520, 370397313, 828990864, 1755830856, 3544407792, 6858847506, 12784761192, 23047190190, 40318901436, 68648343984, 114043745376] This concludes this chapter, that took, 141.574, 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 + 1, r + 1, r, r, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 3526 r + 24190 r + 89780 r 3 2 + 192915 r + 235734 r + 149160 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+3526*r^6+24190*r^5+89780*r^4+192915* r^3+235734*r^2+149160*r+50400) The first few terms, starting at, 1, are [74, 1577, 16680, 114562, 586002, 2417721, 8464962, 26033733, 72078292, 182937326, 431533466, 956311980, 2007969484, 4022303190, 7730383548, 14320813209, 25673185206, 44687102383, 75737384756, 125293495118] ----------------------------------------------------------------- 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 + 2, r, r, r, r], is given by 4 3 2 r (r + 5) (r + 4) (r + 3) (r + 1) (215 r + 2021 r + 6943 r + 10399 r + 5622) 2 (r + 2) /1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^4+2021*r^3+6943*r^2+10399*r+5622)*(r +2)^2 The first few terms, starting at, 1, are [45, 1230, 14274, 102807, 540918, 2272908, 8058798, 25010271, 69713787, 177853104, 421241964, 936525954, 1971584316, 3957938640, 7620343908, 14138293338, 25378514073, 44222775234, 75021589566, 124211788305] 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 + 1, r + 1, r, r, r], is given by 4 3 2 r (r + 5) (r + 4) (r + 3) (r + 1) (215 r + 2021 r + 6943 r + 10399 r + 5622) 2 (r + 2) /1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+1)*(215*r^4+2021*r^3+6943*r^2+10399*r+5622)*(r +2)^2 The first few terms, starting at, 1, are [45, 1230, 14274, 102807, 540918, 2272908, 8058798, 25010271, 69713787, 177853104, 421241964, 936525954, 1971584316, 3957938640, 7620343908, 14138293338, 25378514073, 44222775234, 75021589566, 124211788305] ----------------------------------------------------------------- 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 + 2, r, r, r, r], is given by r (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2451 r + 9415 r + 13365 r + 2490 r - 7776)/1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2451*r^4+9415*r^3+13365*r^2+ 2490*r-7776) The first few terms, starting at, 1, are [12, 717, 10416, 83310, 464928, 2026827, 7365732, 23260380, 65667888, 169152984 , 403637520, 902701020, 1909429392, 3848079234, 7432687368, 13827304416, 24876869556, 43432980327, 73805062848, 122374837530] This concludes this chapter, that took, 286.084, 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 + 1, r + 1, r + 1, r, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 3999 r + 31285 r + 133345 r 3 2 + 332940 r + 481616 r + 367320 r + 100800)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+3999*r^6+31285*r^5+133345*r^4+332940 *r^3+481616*r^2+367320*r+100800) The first few terms, starting at, 1, are [144, 2632, 25219, 162026, 789957, 3143370, 10698666, 32163780, 87402029, 218389314, 508369303, 1113801052, 2315562758, 4598266284, 8769539148, 16134840318, 28747896990, 49762292228, 83916215691, 138190036534] ----------------------------------------------------------------- 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 + 2, r + 1, r, r, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2924 r + 15715 r + 42340 r + 58770 r + 37956 r + 10080)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2924*r^5+15715*r^4+42340*r^3+ 58770*r^2+37956*r+10080) The first few terms, starting at, 1, are [100, 2122, 21868, 146373, 732009, 2962377, 10202346, 30936048, 84608953, 212461249, 496502656, 1091205622, 2274359790, 4525917990, 8646662196, 15932223387, 28422505650, 49252002844, 83132973484, 137011081075] ----------------------------------------------------------------- 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 + 3, r, r, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3999 r + 28435 r + 97225 r + 170190 r + 139856 r + 43920)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3999*r^5+28435*r^4+97225*r^3+ 170190*r^2+139856*r+43920) The first few terms, starting at, 1, are [48, 1416, 16775, 121132, 634740, 2649738, 9326559, 28733892, 79534026, 201577662, 474529770, 1049068904, 2197061776, 4389495132, 8413937466, 15547000344, 27801776676, 48275661852, 81630431025, 134744076852] 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 + 1, r + 1, r + 1, r, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 2924 r + 15715 r + 42340 r + 58770 r + 37956 r + 10080)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+2924*r^5+15715*r^4+42340*r^3+ 58770*r^2+37956*r+10080) The first few terms, starting at, 1, are [100, 2122, 21868, 146373, 732009, 2962377, 10202346, 30936048, 84608953, 212461249, 496502656, 1091205622, 2274359790, 4525917990, 8646662196, 15932223387, 28422505650, 49252002844, 83132973484, 137011081075] ----------------------------------------------------------------- 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 + 2, r + 1, r, r, r], is given by r (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2924 r + 14145 r + 28780 r + 23320 r + 11256)/1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2924*r^4+14145*r^3+28780*r^2 +23320*r+11256) The first few terms, starting at, 1, are [48, 1416, 16854, 121950, 639357, 2668578, 9388764, 28910310, 79979757, 202605546, 476730540, 1053499044, 2205526518, 4404961596, 8441121792, 15593182884, 27877911006, 48397856628, 81821898642, 135037659570] ----------------------------------------------------------------- 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 + 3, r, r, r, r], is given by r (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2924 r + 11935 r + 10810 r - 2340 r - 3384)/1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2924*r^4+11935*r^3+10810*r^2 -2340*r-3384) The first few terms, starting at, 1, are [12, 768, 11637, 94638, 530802, 2313036, 8380713, 26355186, 74059128, 189859956 , 450929622, 1003927860, 2114468772, 4244103720, 8166536262, 15138473820, 27145019484, 47244926520, 80047495539, 132360514770] 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 + 1, r + 1, r + 1, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3999 r + 28435 r + 97225 r + 170190 r + 139856 r + 43920)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3999*r^5+28435*r^4+97225*r^3+ 170190*r^2+139856*r+43920) The first few terms, starting at, 1, are [48, 1416, 16775, 121132, 634740, 2649738, 9326559, 28733892, 79534026, 201577662, 474529770, 1049068904, 2197061776, 4389495132, 8413937466, 15547000344, 27801776676, 48275661852, 81630431025, 134744076852] ----------------------------------------------------------------- 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 + 2, r + 1, r, r, r], is given by r (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 5 4 3 2 (215 r + 2924 r + 11935 r + 10810 r - 2340 r - 3384)/1209600 and in Maple input form 1/1209600*r*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^5+2924*r^4+11935*r^3+10810*r^2 -2340*r-3384) The first few terms, starting at, 1, are [12, 768, 11637, 94638, 530802, 2313036, 8380713, 26355186, 74059128, 189859956 , 450929622, 1003927860, 2114468772, 4244103720, 8166536262, 15138473820, 27145019484, 47244926520, 80047495539, 132360514770] ----------------------------------------------------------------- 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 + 3, r, r, r, r], is given by 8 7 6 5 4 r (r + 2) (r + 1) (215 r + 5504 r + 54298 r + 245708 r + 425155 r 3 2 - 155404 r - 757588 r - 91488 r + 475200)/1209600 and in Maple input form 1/1209600*r*(r+2)*(r+1)*(215*r^8+5504*r^7+54298*r^6+245708*r^5+425155*r^4-\ 155404*r^3-757588*r^2-91488*r+475200) The first few terms, starting at, 1, are [1, 296, 6764, 65988, 409306, 1898868, 7175058, 23242980, 66752928, 173979784, 418550132, 941371652, 1999062273, 4039538920, 7816396884, 14557387368, 26206786203, 45766851768, 77770012564, 128921029556] This concludes this chapter, that took, 545.419, 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 + 1, r + 1, r + 1, r + 1, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 22845 r + 83015 r + 172060 r + 192228 r + 90720)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+22845*r^4+83015*r^3+ 172060*r^2+192228*r+90720) The first few terms, starting at, 1, are [336, 4824, 40500, 239163, 1099584, 4190208, 13795584, 40396356, 107450772, 263770650, 604919016, 1308569730, 2690737968, 5292298152, 10008343632, 18276808035, 32347466712, 55658080236, 93351149628, 152972651205] ----------------------------------------------------------------- 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 + 2, r + 1, r + 1, r, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 21895 r + 73475 r + 133650 r + 121848 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+21895*r^4+73475*r^3+ 133650*r^2+121848*r+40320) The first few terms, starting at, 1, are [235, 3915, 35268, 216660, 1020864, 3954291, 13168716, 38883702, 104077831, 256729330, 591018168, 1282411728, 2643521352, 5210121138, 9869854152, 18050012616, 31985472651, 55093502893, 92488868956, 151680548580] ----------------------------------------------------------------- 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 + 2, r + 2, r, r, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 20945 r + 63935 r + 99560 r + 70668 r + 20160)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+20945*r^4+63935*r^3+ 99560*r^2+70668*r+20160) The first few terms, starting at, 1, are [166, 3185, 30740, 196359, 948000, 3732168, 12571416, 37429821, 100814714, 249882919, 577448456, 1256795618, 2597163168, 5129265840, 9733349184, 17826130980, 31627671966, 54534852581, 91634830012, 150399737345] ----------------------------------------------------------------- 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 + 3, r + 1, r, r, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36980 r + 155540 r 3 2 + 355625 r + 428588 r + 238260 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36980*r^5+155540*r^4+355625 *r^3+428588*r^2+238260*r+50400) The first few terms, starting at, 1, are [126, 2700, 27458, 180759, 889641, 3548718, 12066351, 36177372, 97961149, 243821292, 565309251, 1233677978, 2555010342, 5055263820, 9607698954, 17619012075, 31295177856, 54013636468, 90835152672, 149196556377] ----------------------------------------------------------------- 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 + 4, r, r, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 34130 r + 116180 r + 189995 r + 129908 r + 39180)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+34130*r^4+116180*r^3+ 189995*r^2+129908*r+39180) The first few terms, starting at, 1, are [51, 1602, 19341, 140175, 732834, 3044778, 10656690, 32640003, 89827881, 226420194, 530260731, 1166619714, 2432263236, 4839077268, 9239628180, 17010882210, 30316977999, 52477561290, 88474868361, 145640611017] 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 + 1, r + 1, r + 1, r + 1, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 21895 r + 73475 r + 133650 r + 121848 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+21895*r^4+73475*r^3+ 133650*r^2+121848*r+40320) The first few terms, starting at, 1, are [235, 3915, 35268, 216660, 1020864, 3954291, 13168716, 38883702, 104077831, 256729330, 591018168, 1282411728, 2643521352, 5210121138, 9869854152, 18050012616, 31985472651, 55093502893, 92488868956, 151680548580] ----------------------------------------------------------------- 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 + 2, r + 1, r + 1, r, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 20325 r + 57275 r + 77080 r + 43308 r + 10080)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+20325*r^4+57275*r^3+ 77080*r^2+43308*r+10080) The first few terms, starting at, 1, are [126, 2731, 27826, 183057, 899778, 3584328, 12172578, 36457278, 98629960, 245297481, 568360156, 1239643366, 2566134156, 5075172384, 9642073260, 17676512535, 31388692170, 54161935135, 91065052254, 149545708015] ----------------------------------------------------------------- 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 + 2, r + 2, r, r, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36360 r + 144610 r 3 2 + 291265 r + 305958 r + 204960 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36360*r^5+144610*r^4+291265 *r^3+305958*r^2+204960*r+50400) The first few terms, starting at, 1, are [103, 2323, 24696, 167360, 839526, 3392457, 11640882, 35134803, 95613661, 238890652, 555539062, 1215254352, 2521721996, 4997315478, 9510072204, 17459245293, 31040411283, 53616739825, 90229720636, 148290521632] ----------------------------------------------------------------- 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 + 3, r + 1, r, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 35100 r + 125770 r + 197965 r + 123438 r + 37200)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+35100*r^4+125770*r^3+ 197965*r^2+123438*r+37200) The first few terms, starting at, 1, are [52, 1667, 20149, 145518, 757458, 3134145, 10929930, 33373824, 91607802, 230396452, 538559879, 1182979564, 2462978700, 4894368102, 9335566932, 17172052104, 30580066344, 52896134577, 89125620793, 146631429098] ----------------------------------------------------------------- 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 + 4, r, r, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 31940 r + 80270 r + 23165 r - 7702 r - 11400)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+31940*r^4+80270*r^3+23165 *r^2-7702*r-11400) The first few terms, starting at, 1, are [12, 819, 12858, 106124, 598248, 2607855, 9430014, 29561202, 82760964, 211341702, 499989204, 1108904160, 2326995632, 4654334970, 8926183812, 16494740484, 29489410332, 51182009373, 86489976342, 142658522028] 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 + 1, r + 1, r + 1, r + 1, r], is given by (r + 5) (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 3397 r + 20945 r + 63935 r + 99560 r + 70668 r + 20160)/ 1209600 and in Maple input form 1/1209600*(r+5)*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+3397*r^5+20945*r^4+63935*r^3+ 99560*r^2+70668*r+20160) The first few terms, starting at, 1, are [166, 3185, 30740, 196359, 948000, 3732168, 12571416, 37429821, 100814714, 249882919, 577448456, 1256795618, 2597163168, 5129265840, 9733349184, 17826130980, 31627671966, 54534852581, 91634830012, 150399737345] ----------------------------------------------------------------- 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 + 2, r + 1, r + 1, r, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36360 r + 144610 r 3 2 + 291265 r + 305958 r + 204960 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36360*r^5+144610*r^4+291265 *r^3+305958*r^2+204960*r+50400) The first few terms, starting at, 1, are [103, 2323, 24696, 167360, 839526, 3392457, 11640882, 35134803, 95613661, 238890652, 555539062, 1215254352, 2521721996, 4997315478, 9510072204, 17459245293, 31040411283, 53616739825, 90229720636, 148290521632] ----------------------------------------------------------------- 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 + 2, r + 2, r, r, r], is given by 9 8 7 6 5 (r + 3) (r + 1) (215 r + 5762 r + 63342 r + 365076 r + 1178895 r 4 3 2 + 2224338 r + 2940268 r + 3076984 r + 1233120 r + 403200)/1209600 and in Maple input form 1/1209600*(r+3)*(r+1)*(215*r^9+5762*r^8+63342*r^7+365076*r^6+1178895*r^5+ 2224338*r^4+2940268*r^3+3076984*r^2+1233120*r+403200) The first few terms, starting at, 1, are [76, 1799, 20328, 144285, 747972, 3094419, 10802324, 33026268, 90765916, 228531589, 534713048, 1175495100, 2449114344, 4869739743, 9293376552, 17102036363, 30467090652, 52718343331, 88852038616, 146218890311] ----------------------------------------------------------------- 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 + 3, r + 1, r, r, r], is given by 7 6 5 4 3 r (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 51728 r + 242380 r + 552125 r 2 + 645448 r + 813252 r + 108720)/1209600 and in Maple input form 1/1209600*r*(r+3)*(r+2)*(r+1)*(215*r^7+5332*r^6+51728*r^5+242380*r^4+552125*r^3 +645448*r^2+813252*r+108720) The first few terms, starting at, 1, are [48, 1452, 17805, 131532, 698019, 2932002, 10343835, 31867671, 88087725, 222777984, 523087422, 1153196499, 2408218098, 4797600804, 9170405370, 16898664258, 30139704294, 52203925836, 88061207619, 145026970902] ----------------------------------------------------------------- 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 + 4, r, r, r, r], is given by 8 7 6 5 4 r (r + 3) (r + 1) (215 r + 5762 r + 59542 r + 288116 r + 636055 r 3 2 + 618098 r + 719708 r - 102696 r - 410400)/1209600 and in Maple input form 1/1209600*r*(r+3)*(r+1)*(215*r^8+5762*r^7+59542*r^6+288116*r^5+636055*r^4+ 618098*r^3+719708*r^2-102696*r-410400) The first few terms, starting at, 1, are [12, 774, 11988, 99594, 566808, 2492253, 9076584, 28617699, 80492184, 206322402 , 489617040, 1088657505, 2289339936, 4587156495, 8810600832, 16302109098, 29177304612, 50688913374, 85728378252, 141506048148] 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 + 1, r + 1, r + 1, r + 1, r], is given by 7 6 5 4 (r + 4) (r + 3) (r + 2) (r + 1) (215 r + 4472 r + 36980 r + 155540 r 3 2 + 355625 r + 428588 r + 238260 r + 50400)/1209600 and in Maple input form 1/1209600*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^7+4472*r^6+36980*r^5+155540*r^4+355625 *r^3+428588*r^2+238260*r+50400) The first few terms, starting at, 1, are [126, 2700, 27458, 180759, 889641, 3548718, 12066351, 36177372, 97961149, 243821292, 565309251, 1233677978, 2555010342, 5055263820, 9607698954, 17619012075, 31295177856, 54013636468, 90835152672, 149196556377] ----------------------------------------------------------------- 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 + 2, r + 1, r + 1, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 35100 r + 125770 r + 197965 r + 123438 r + 37200)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+35100*r^4+125770*r^3+ 197965*r^2+123438*r+37200) The first few terms, starting at, 1, are [52, 1667, 20149, 145518, 757458, 3134145, 10929930, 33373824, 91607802, 230396452, 538559879, 1182979564, 2462978700, 4894368102, 9335566932, 17172052104, 30580066344, 52896134577, 89125620793, 146631429098] ----------------------------------------------------------------- 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 + 2, r + 2, r, r, r], is given by 7 6 5 4 3 r (r + 3) (r + 2) (r + 1) (215 r + 5332 r + 51728 r + 242380 r + 552125 r 2 + 645448 r + 813252 r + 108720)/1209600 and in Maple input form 1/1209600*r*(r+3)*(r+2)*(r+1)*(215*r^7+5332*r^6+51728*r^5+242380*r^4+552125*r^3 +645448*r^2+813252*r+108720) The first few terms, starting at, 1, are [48, 1452, 17805, 131532, 698019, 2932002, 10343835, 31867671, 88087725, 222777984, 523087422, 1153196499, 2408218098, 4797600804, 9170405370, 16898664258, 30139704294, 52203925836, 88061207619, 145026970902] ----------------------------------------------------------------- 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 + 3, r + 1, r, r, r], is given by 8 7 6 5 4 r (r + 2) (r + 1) (215 r + 5977 r + 66154 r + 361594 r + 966535 r 3 2 + 1033873 r + 294056 r + 50076 r - 359280)/1209600 and in Maple input form 1/1209600*r*(r+2)*(r+1)*(215*r^8+5977*r^7+66154*r^6+361594*r^5+966535*r^4+ 1033873*r^3+294056*r^2+50076*r-359280) The first few terms, starting at, 1, are [12, 819, 12858, 106290, 599850, 2616390, 9463200, 29666367, 83048856, 212047011, 501572214, 1112213037, 2333515002, 4666553748, 8948123808, 16532697234, 29552968980, 51285411363, 86653927266, 142912547316] ----------------------------------------------------------------- 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 + 4, r, r, r, r], is given by 8 7 6 5 4 r (r + 2) (r + 1) (215 r + 5977 r + 62684 r + 285244 r + 390545 r 3 2 - 549617 r - 276324 r + 282156 r + 720)/1209600 and in Maple input form 1/1209600*r*(r+2)*(r+1)*(215*r^8+5977*r^7+62684*r^6+285244*r^5+390545*r^4-\ 549617*r^3-276324*r^2+282156*r+720) The first few terms, starting at, 1, are [1, 303, 7175, 71664, 449940, 2098920, 7945071, 25728411, 73776054, 191855521, 460359900, 1032554315, 2186542683, 4406079120, 8502440418, 15793392666, 28359931335, 49407062091, 83761538437, 138547308306] 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 + 1, r + 1, r + 1, r + 1, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 34130 r + 116180 r + 189995 r + 129908 r + 39180)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+34130*r^4+116180*r^3+ 189995*r^2+129908*r+39180) The first few terms, starting at, 1, are [51, 1602, 19341, 140175, 732834, 3044778, 10656690, 32640003, 89827881, 226420194, 530260731, 1166619714, 2432263236, 4839077268, 9239628180, 17010882210, 30316977999, 52477561290, 88474868361, 145640611017] ----------------------------------------------------------------- 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 + 2, r + 1, r + 1, r, r], is given by r (r + 4) (r + 3) (r + 2) (r + 1) 6 5 4 3 2 (215 r + 4472 r + 31940 r + 80270 r + 23165 r - 7702 r - 11400)/ 1209600 and in Maple input form 1/1209600*r*(r+4)*(r+3)*(r+2)*(r+1)*(215*r^6+4472*r^5+31940*r^4+80270*r^3+23165 *r^2-7702*r-11400) The first few terms, starting at, 1, are [12, 819, 12858, 106124, 598248, 2607855, 9430014, 29561202, 82760964, 211341702, 499989204, 1108904160, 2326995632, 4654334970, 8926183812, 16494740484, 29489410332, 51182009373, 86489976342, 142658522028] ----------------------------------------------------------------- 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 + 2, r + 2, r, r, r], is given by 8 7 6 5 4 r (r + 3) (r + 1) (215 r + 5762 r + 59542 r + 288116 r + 636055 r 3 2 + 618098 r + 719708 r - 102696 r - 410400)/1209600 and in Maple input form 1/1209600*r*(r+3)*(r+1)*(215*r^8+5762*r^7+59542*r^6+288116*r^5+636055*r^4+ 618098*r^3+719708*r^2-102696*r-410400) The first few terms, starting at, 1, are [12, 774, 11988, 99594, 566808, 2492253, 9076584, 28617699, 80492184, 206322402 , 489617040, 1088657505, 2289339936, 4587156495, 8810600832, 16302109098, 29177304612, 50688913374, 85728378252, 141506048148] ----------------------------------------------------------------- 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 + 3, r + 1, r, r, r], is given by 8 7 6 5 4 r (r + 2) (r + 1) (215 r + 5977 r + 62684 r + 285244 r + 390545 r 3 2 - 549617 r - 276324 r + 282156 r + 720)/1209600 and in Maple input form 1/1209600*r*(r+2)*(r+1)*(215*r^8+5977*r^7+62684*r^6+285244*r^5+390545*r^4-\ 549617*r^3-276324*r^2+282156*r+720) The first few terms, starting at, 1, are [1, 303, 7175, 71664, 449940, 2098920, 7945071, 25728411, 73776054, 191855521, 460359900, 1032554315, 2186542683, 4406079120, 8502440418, 15793392666, 28359931335, 49407062091, 83761538437, 138547308306] ----------------------------------------------------------------- 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 + 4, r, r, r, r], is given by 8 7 6 5 4 r (r - 1) (r + 1) (215 r + 6622 r + 77170 r + 387472 r + 518395 r 3 2 - 1422182 r - 1004740 r + 1484568 r + 1617120)/1209600 and in Maple input form 1/1209600*r*(r-1)*(r+1)*(215*r^8+6622*r^7+77170*r^6+387472*r^5+518395*r^4-\ 1422182*r^3-1004740*r^2+1484568*r+1617120) The first few terms, starting at, 1, are [0, 78, 3312, 43086, 312312, 1591542, 6389940, 21568815, 63760128, 169677420, 414506400, 943026513, 2020029804, 4109053026, 7991514024, 14942141874, 26981217072, 47229705354, 80399995680, 133462765824] This concludes this chapter, that took, 1109.256, seconds to produce. This ends this wonderful webbook that took, 2193.518, seconds.