An involution on \beta(1,0)-trees

TL;DR

This paper proves that a previously introduced transformation on -0-trees is an involution and refines related statistical distributions, enhancing the theoretical understanding of these combinatorial structures.

Contribution

It provides the missing proof that the transformation is an involution and refines the equidistribution results for -0-trees.

Findings

Confirmed the involution property of h on -0-trees.

Refined the equidistribution results for statistics on -0-trees.

Abstract

In [Decompositions and statistics for \beta(1,0)-trees and nonseparable permutations, Advances Appl. Math. 42 (2009) 313--328] we introduced an involution, h, on \beta(1,0)-trees. We neglected, however, to prove that h indeed is an involution. In this note we provide the missing proof. We also refine an equidistribution result given in the same paper.