The Gessel Correspondence and the Partial $\gamma$-Positivity of the Eulerian Polynomials on Multiset Stirling Permutations

Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we come to the realization that the grammatical labels play a role as records of chopped off leaves of the original increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial $\gamma$-positivity of the Eulearian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels $x$ and $y$ in the Gessel representation of a $k$-Stirling permutation by means of an increasing $(k+1)$-ary tree. More precisely, we introduce a Foata-Strehl action on the Gessel trees resulting in an interpretation of the partial $\gamma$-coefficients of the aforementioned Eulerian polynomials, different from the ones found by Lin-Ma-Zhang and Yan-Huang-Yang. In particular, our strategy can be adapted to deal with the partial $\gamma$-coefficients of the second order Eulerian polynomials, which in turn can be readily converted to the combinatorial formulation due to Ma-Ma-Yeh in connection with certain statistics of Stirling permutations.