Markov bases for noncommutative Fourier analysis of ranked data

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.