Equidistributions around special kinds of descents and excedances

TL;DR

This paper introduces refined Eulerian polynomials through continued fractions, providing combinatorial interpretations of permutation statistics, and proves equidistribution results that confirm and extend recent conjectures in permutation combinatorics.

Contribution

It develops new polynomial refinements and combinatorial interpretations of permutation statistics, and establishes equidistribution results that support recent conjectures.

Findings

Derived combinatorial interpretations for refined Eulerian polynomials.

Proved equidistribution of certain permutation statistics.

Confirmed and extended recent conjectures in permutation statistics.

Abstract

We consider a sequence of four variable polynomials by refining Stieltjes' continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a by-product, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several compagnion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov.