Catalan-Spitzer permutations

We study two classes of permutations intimately related to the visual proof of Spitzer’s lemma and Huq’s generalization of the Chung-Feller theorem. Both classes of permutations are counted by the Fuss-Catalan numbers. The study of one class leads to a generalization of results of Flajolet from continued fractions to continuants. The study of the other class leads to the discovery of a restricted variant of the Foata–Strehl group action.