  To calibrate Fourier analysis of $S_5$ ranking data by Markov chain Monte
Carlo techniques, a set of moves (Markov basis) is needed. We calculate this
basis, and use it to provide a new statistical analysis of two data sets. The
calculation involves a large Gr\"obner basis computation (45825 generators),
but reduction to a minimal basis and reduction by natural symmetries leads to a
remarkably small basis (14 elements). Although the Gr\"obner basis calculation
is infeasible for $S_6$, we exploit the symmetry of the problem to calculate a
Markov basis for $S_6$ with 7,113,390 elements in 58 symmetry classes. We
improve a bound on the degree of the generators for a Markov basis for $S_n$
and conjecture that this ideal is generated in degree 3.
