New refinements of Narayana polynomials and Motzkin polynomials
Janet J.W. Dong, Lora R. Du, Kathy Q. Ji, Dax T.X. Zhang

TL;DR
This paper introduces new polynomial refinements of Narayana and Motzkin polynomials related to plane trees, providing generating functions and a unified grammatical approach to analyze their combinatorial properties.
Contribution
It develops a new polynomial $G_n$ refining Narayana polynomials and a refined Motzkin polynomial $M_n$, along with their generating functions and a grammatical method for analysis.
Findings
Derived generating functions for $G_n$ and $M_n$ polynomials.
Established a refined Coker's formula connecting these polynomials.
Simplified derivations using a grammatical approach.
Abstract
Chen, Deutsch and Elizalde introduced a refinement of the Narayana polynomials by distinguishing between old (leftmost child) and young leaves of plane trees. They also provided a refinement of Coker's formula by constructing a bijection. In fact, Coker's formula establishes a connection between the Narayana polynomials and the Motzkin polynomials, which implies the -positivity of the Narayana polynomials. In this paper, we introduce the polynomial , which further refine the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for . To achieve further refinement of Coker's formula based on the polynomial , we consider a refinement of the Motzkin polynomials by…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Algebraic structures and combinatorial models
