Probability calculations for transitivity of simple majority rule with anonymous voters

Let P(n, 4) denote the probability that there is a pairwise simple majority rule winner on four candidates with n voters under the impartial anonymous culture condition. Similarly, let P^t(n, 4) denote the probability that the complete ranking obtained by pairwise simple majority voting on four candidates with n voters is transitive under the impartial anonymous culture condition. We obtain representations for P(n, 4) and P^t(n, 4), and computed values for each for odd n with 3 n 19.This research was supported through a fellowship from the Center for Advanced Study, University of Delaware.