Labeled Plane Trees and Increasing Plane Trees
The main aim of this paper is to establish a polynomial analogue of $(n+1)!C_n=2^n(2n-1)!!$ (with $C_n$ as the $n$-th Catalan number) in the setting of labeled plane trees and increasing plane trees. This analogue is formulated in terms of improper edges of labeled plane trees and yields explicit formulas for the generating polynomials defined on labeled plane trees refined by improper and proper edges, together with a root-degree refinement for trees rooted at $0$. To prove this result, we construct a new involution on labeled plane trees, which implies that the number of improper edges and the number of proper edges are equidistributed over the set of labeled plane trees. We further apply this involution to establish pairwise symmetry properties of multivariable polynomials defined on labeled plane trees involving several classes of leaves and interior vertices. More precisely, certain specializations of these polynomials are invariant under the subgroup of $S_6$ generated by the three disjoint transpositions $(12)$, $(34)$, and $(56)$. As special cases, our results recover the symmetry properties for plane trees and tip-augmented plane trees due to Dong, Du, Ji and Zhang. Finally, via the Koganov--Janson correspondence, improper edges of labeled plane trees correspond bijectively to improper arcs of quasi-Stirling permutations, leading to an explicit formula for the generating function defined on quasi-Stirling permutations refined by improper arcs.