Catalan-Spitzer permutations

TL;DR

This paper explores two classes of permutations linked to Spitzer's lemma and Chung-Feller theorem, revealing their enumeration by Fuss-Catalan numbers and uncovering new combinatorial structures and generalizations.

Contribution

It introduces two permutation classes related to classical theorems, extending known results and discovering new algebraic and combinatorial structures.

Findings

Both classes are counted by Fuss-Catalan numbers.

One class generalizes Flajolet's continued fraction results.

The other class reveals a restricted Foata--Strehl group action.

Abstract

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.