New refinements of Narayana polynomials and Motzkin polynomials

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 Gₙ(x₁₁,x₁₂,x₂;y₁₁,y₁₂,y₂), which further refine the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for Gₙ(x₁₁,x₁₂,x₂;y₁₁,y₁₂,y₂). To achieve further refinement of Coker's formula based on the polynomial Gₙ(x₁₁,x₁₂,x₂;y₁₁,y₁₂,y₂), we consider a refinement Mₙ(u₁,u₂,u₃;v₁,v₂) 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 Mₙ(u₁,u₂,u₃;v₁,v₂) is also established, and the refinement of Coker's formula is immediately derived by combining the generating function for Gₙ(x₁₁,x₁₂,x₂;y₁₁,y₁₂,y₂) and the generating function for Mₙ(u₁,u₂,u₃;v₁,v₂). 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 Gₙ(x₁₁,x₁₂,x₂;y₁₁,y₁₂,y₂) and Mₙ(u₁,u₂,u₃;v₁,v₂) become quite simple once their grammars are established.