A Context-free Grammar for the $e$-Positivity of the Trivariate Second-order Eulerian Polynomials

Abstract

Ma-Ma-Yeh made a beautiful observation that a change of the grammar of Dumont instantly leads to the γ-positivity of the Eulearian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the e-positivity of the trivariate second-order Eulerian polynomials first introduced by Dumont in the context of ternary trees, and independently defined by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.