New refinements of Narayana polynomials and Motzkin polynomials

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 $\gamma$-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.