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

Symmetric generating functions and Euler-Stirling statistics on permutations