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

We prove that the enumerative polynomials of generalized Stirling permutations by the statistics of plateaux, descents and ascents are partial $γ$-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial $γ$-positivity conjecture due to Ma, Yeh and the second named author. Our partial $γ$-positivity expansion, as well as a combinatorial interpretation for the corresponding $γ$-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 $γ$-positivity from the stability of certain multivariate polynomials.

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