Spectral criterion for testing hypotheses on random permutations

Suppose that for each of $N$ independent identically distributed random permutations we observe a pair consisting of a random uniformly distributed argument and a corresponding value of permutation. We consider the problem of testing the hypothesis that the distribution of permutations is uniform against the hypothesis that permutations are the products of r independent permutations with known distribution. A test constructed by eigenvectors of matrices of transition probabilities (arguments to values) is proposed and investigated.