Plateaux on generalized Stirling permutations and partial $γ$-positivity

Abstract: We prove that the enumerative polynomials of generalized Stirling permutations by the statistics of plateaux, descents and ascents are partial $\gamma$-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial $\gamma$-positivity conjecture due to Ma, Yeh and the second named author. Our partial $\gamma$-positivity expansion, as well as a combinatorial interpretation for the corresponding $\gamma$-coefficients, are obtained via the machine of context-free grammars and a group action on generalized Stirling permutations. Besides, we also provide an alternative approach to the partial $\gamma$-positivity from the stability of certain multivariate polynomials.