Symmetric generating functions and Euler-Stirling statistics on permutations

We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijections on the multiple equidistributions of Euler-Stirling statistics over permutations and transformation formulas of basic hypergeometric series. Furthermore, we establish a new quadruple equidistribution of Euler-Stirling statistics over inversion sequences, as progress towards a conjecture proposed by Schlosser and the author (2020).