Partial $\gamma$-Positivity for Quasi-Stirling Permutations

We prove that the enumerative polynomials of quasi-Stirling permutations with respect to the statistics of plateaux, descents and ascents are partial $\gamma$-positive, thereby confirming a recent conjecture posed by Lin, Ma and Zhang. This is accomplished by proving the partial $\gamma$-positivity of the enumerative polynomials of certain ordered labeled trees, which are in bijection with quasi-Stirling permutations. As an application, we provide an alternative proof of the partial $\gamma$-positivity of the enumerative polynomials on Stirling permutations.