On the Most Commonly-Occuring Score Vectors for American Tournaments of n-players, and their Corresponding Records



                              By  Shalosh B. Ekhad



In his beautiful article, "An American Tournament Treated by the Calculus of\
     Symmetric Functions"

     Quarterly J. of Pure and Applied Mathematics. vol. XLIX, No. 193, 1920

The Combinatorial Giant, (and amazing human calculator), Major Percy Alexand\
    er MacMahon

considered round-robin tournaments for n players for n from 2 to 9, and for \
    each found all the

possible score vectors, and determined the most popular one, as well as the \
    number of times it shows up



In this article, we continue this work up to n=, 12,

    and thereby continue OEIS sequence A274098

currently (June 13, 2016) only containing the terms MacMahon computed purely\
     by hand





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                       In a torunament with, 2, players

                   The sole  champion score-vector is, [1, 0]

                         the corresponding record is, 2



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                       In a torunament with, 3, players

                 The sole  champion score-vector is, [2, 1, 0]

                         the corresponding record is, 6



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                       In a torunament with, 4, players

       The set of champion score vectors is, {[2, 2, 1, 1], [3, 2, 1, 0]}

                        the corresponding record is, 24



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                       In a torunament with, 5, players

              The sole  champion score-vector is, [3, 2, 2, 2, 1]

                        the corresponding record is, 280



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                       In a torunament with, 6, players

             The sole  champion score-vector is, [4, 3, 3, 2, 2, 1]

                       the corresponding record is, 8640



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                       In a torunament with, 7, players

           The sole  champion score-vector is, [4, 4, 3, 3, 3, 2, 2]

                      the corresponding record is, 233520



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                       In a torunament with, 8, players

          The sole  champion score-vector is, [5, 4, 4, 4, 3, 3, 3, 2]

                     the corresponding record is, 23157120



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                       In a torunament with, 9, players

        The sole  champion score-vector is, [6, 5, 5, 4, 4, 4, 3, 3, 2]

                    the corresponding record is, 5329376640



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                       In a torunament with, 10, players

       The sole  champion score-vector is, [7, 6, 6, 5, 5, 4, 4, 3, 3, 2]

                   the corresponding record is, 1314169920000



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                       In a torunament with, 11, players

     The sole  champion score-vector is, [7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3]

                 the corresponding record is, 1016970317932800



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                       In a torunament with, 12, players

    The sole  champion score-vector is, [8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 3]

                the corresponding record is, 1772428331094220800



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     In conclusion, the n=2 thorough n=, 12, of OEIS sequence A274098 are



[2, 6, 24, 280, 8640, 233520, 23157120, 5329376640, 1314169920000, 1016970317932800, 1772428331094220800]


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        This ends this article, that took, 10.935, seconds to produce.