New refinements of Narayana polynomials and Motzkin polynomials

Janet J. W. Dong; Lora R. Du; Kathy Q. Ji; Dax T. X. Zhang

paper

New refinements of Narayana polynomials and Motzkin polynomials

arXiv:2408.06912

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 $γ$ -positivity of the Narayana polynomials. In this paper, we introduce the polynomial $G_{n} (x_{11}, x_{12}, x_{2}; y_{11}, y_{12}, y_{2})$ , which further refine the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for $G_{n} (x_{11}, x_{12}, x_{2}; y_{11}, y_{12}, y_{2})$ . To achieve further refinement of Coker's formula based on the polynomial $G_{n} (x_{11}, x_{12}, x_{2}; y_{11}, y_{12}, y_{2})$ , we consider a refinement $M_{n} (u_{1}, u_{2}, u_{3}; v_{1}, v_{2})$ of the Motzkin polynomials by classifying the old leaves of a tip-augmented plane tree into three categories and the young leaves into two categories. The generating function for $M_{n} (u_{1}, u_{2}, u_{3}; v_{1}, v_{2})$ is also established, and the refinement of Coker's formula is immediately derived by combining the generating function for $G_{n} (x_{11}, x_{12}, x_{2}; y_{11}, y_{12}, y_{2})$ and the generating function for $M_{n} (u_{1}, u_{2}, u_{3}; v_{1}, v_{2})$ . We derive several interesting consequences from this refinement of Coker's formula. The method used in this paper is the grammatical approach introduced by Chen. We develop a unified grammatical approach to exploring polynomials associated with the statistics defined on plane trees. As you will see, the derivations of the generating functions for $G_{n} (x_{11}, x_{12}, x_{2}; y_{11}, y_{12}, y_{2})$ and $M_{n} (u_{1}, u_{2}, u_{3}; v_{1}, v_{2})$ become quite simple once their grammars are established.

40 pages