Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers

TL;DR

This paper develops new continued fraction representations for multivariate polynomials that count D-permutations, a combinatorial model for Genocchi numbers, based on various permutation statistics.

Contribution

It introduces novel Stieltjes-type and Thron-type continued fractions for multivariate polynomials enumerating D-permutations with multiple statistics.

Findings

Derived continued fractions for polynomials counting D-permutations.

Connected permutation statistics to classical continued fractions.

Provided combinatorial interpretations for the coefficients.

Abstract

A D-permutation is a permutation of $[2n]$ satisfying $2k-1 \le \sigma(2k-1)$ and $2k \ge \sigma(2k)$ for all $k$; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.