How Many  Condorcet Scenarios Are There?



                              By Shalosh B. Ekhad



Suppose that there three candidates and n voters, the sequence for the numbe\
    r of ways (out of binomial(n+5,5)) possible ways

      in which the votes can be cast such that there is a cycle 1>2>3>1 is

[0, 0, 1, 0, 6, 1, 21, 6, 56, 21, 126, 56, 252, 126, 462, 252, 792, 462, 1287,

    792, 2002, 1287, 3003, 2002, 4368, 3003]

hence the number of ways in which there is a cycle 1>2>3>1 or 1>3>2>1, is tw\
    ice that



               Let a(n) be the number of Condorcet scenarios, then



                      infinity
                       -----                3   2
                        \            n   2 t  (t  - t + 1)
                         )     a(n) t  = -----------------
                        /                       5        6
                       -----             (t + 1)  (t - 1)
                       n = 0



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