Equidistributions around special kinds of descents and excedances

Abstract

In a recent paper (arXiv:2101.01928v1) Baril and Kirgizov posed two conjectures on the equidistibution of (cyc, des 2 ) ∼ (cyc, pex) and (des, des 2 ) ∼ (exc, pex), where cyc, des and exc are classical statistics counting the numbers of cycles, descents and excedances of permutation, while des 2 and pex are numbers of special descents and excedances over permutations. In this paper, using combinatorial theory of J-continued fractions and bijections we confirm and strengthen the second conjecture and expand the first one.