The binomial-Stirling–Eulerian polynomials

We introduce the binomial-Stirling–Eulerian polynomials, denoted ${\tilde{A}}_n(x,y|\alpha )$, which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When $\alpha =1$, these polynomials reduce to the binomial-Eulerian polynomials ${\tilde{A}}_n(x,y)$, originally named by Shareshian and Wachs and explored by Chung–Graham–Knuth and Postnikov–Reiner–Williams. We investigate the $\gamma$-positivity of ${\tilde{A}}_n(x,y|\alpha )$ from two aspects: $•$ firstly by employing the grammatical calculus introduced by Chen; $•$ and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the $\gamma$-positivity of ${\tilde{A}}_n(x,y)$ first demonstrated by Postnikov, Reiner and Williams.