On two conjectures concerning special kinds of descents on permutations

In this paper, we prove the continued fraction conjecture posed by Han, Mao and Zeng for the generating function of permutations with respect to the number of descents of type $2$ and the number of cycles, thereby settling their reformulation of a conjecture originally due to Baril and Kirgizov. We further establish the symmetry of the bistatistic $({\text{des}}_2,\text{ear})$ over ${\mathfrak{S}}_n$ as conjectured by Han, Mao and Zeng and strengthen this result by exhibiting five equidistributed companions for $({\text{des}}_2,\text{ear})$. Here the statistic ${\text{des}}_2$ denotes the number of descents of type $2$, and the statistic $\text{ear}$ denotes the number of exclusive antirecord cycle peaks originally introduced by Sokal and Zeng.