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

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