Continued fractions for cycle-alternating permutations

TL;DR

This paper develops continued fractions to enumerate cycle-alternating permutations and their generalizations, providing new combinatorial formulas and interpretations for these structures with respect to various permutation statistics.

Contribution

It introduces new continued fraction representations for multivariate permutation polynomials and generalizes cycle-alternating permutations to Laguerre digraphs with associated generating functions.

Findings

Derived Stieltjes-type continued fractions for cycle-alternating permutations.

Established exponential generating functions for alternating Laguerre digraphs.

Connected continued fractions to combinatorial structures like Laguerre digraphs.

Abstract

A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index $i$ is either a cycle valley ($\sigma^{-1}(i)>i<\sigma(i)$) or a cycle peak ($\sigma^{-1}(i)<i>\sigma(i)$). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes--Rogers and Jacobi--Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs.