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arXiv · 2404.10331

Increasing Binary Trees and the $(\alpha,\beta)$-Eulerian Polynomials

William Y. C. Chen·Amy M. Fu
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Abstract

In light of the grammar given by Ji for the $(\alpha,\beta)$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the $\gamma$-coefficients of the $\alpha$-Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the $\gamma$-coefficients of the derangement polynomials in the same vein. By means of a decomposition of an increasing binary tree into a forest, we find combinatorial interpretations of the sums involving two identities of Ji, one of which can be viewed as $(\alpha,\beta)$-extensions of the formulas of Petersen and Stembridge.

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William Y. C. Chen, Amy M. Fu. 2024-04-16. Increasing Binary Trees and the $(\alpha,\beta)$-Eulerian Polynomials. https://arxiv.org/abs/2404.10331

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