Statistics on Multipermutations and Partial Γ-Positivity

We prove that the enumerative polynomials of Stirling multipermutations 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 Stirling multipermutations. Besides, we also provide an alternative approach to the partial γ-positivity from the stability of certain multivariate polynomials on Stirling multipermutations. Moreover, we prove the partial γ-positivity for the enumerators of multipermutations by plateaux, descents and ascents via introducing a group action on words. Since multipermutations without any plateau are Smirnov words, our result generalizes a γ-positivity result due to Linusson, Shareshian and Wachs in this special case. Interestingly, our second action on multipermutations applies also to Stirling multipermutations and results in another combinatorial expansion for their partial γ-positivity. Finally, using a modification of our second group action and Foata's first fundamental transformation, we prove the partial γ-positivity for the enumerators of multipermutations by fixed points, excedances and drops, generalizing another result of Linusson, Shareshian and Wachs for derangements of a multiset.