Stable multivariate Narayana polynomials and labeled plane trees

In this paper, we introduce stable multivariate generalizations of Narayana polynomials of type A and type B. We give an insertion algorithm for labeled plane trees and introduce the notion of improper edges. Our polynomials are multivariate generating polynomials of labeled plane trees and can be generated by a grammatical labeling based on a context-free grammar. Our proof of real stability uses a characterization of stable-preserving linear operators due to Borcea and Br\"and\'en. In particular, we get an alternative multivariate stable refinement of the second-order Eulerian polynomials, which is different from the one given by Haglund and Visontai.