New refinements of Narayana polynomials and Motzkin polynomials

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 refines 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, including the new symmetries of plane trees, Euler transformation of the Narayana polynomials and the Motzkin polynomials due to Lin and Kim and the real-rootedness of the Motzkin polynomials. 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. 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.