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

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 نشر من قبل William Y. C. Chen
 تاريخ النشر 2021
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Ma-Ma-Yeh made a beautiful observation that a change of the grammar of Dumont instantly leads to the $gamma$-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 introduced by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.



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