Enumeration of fixed points of an involution on $\beta(1,0)$-trees

TL;DR

This paper characterizes and counts fixed points of an involution on $eta(1,0)$-trees, revealing surprising equinumerosity with fixed points under dual map operations on rooted non-separable planar maps.

Contribution

It provides a detailed description and enumeration of fixed points of the involution on $eta(1,0)$-trees, linking them to fixed points under dual map operations.

Findings

Fixed points are explicitly described and enumerated.

Fixed points are equinumerous with those under the dual map operation.

The fixed points do not correspond under the natural bijection between trees and maps.

Abstract

$\beta(1,0)$-trees provide a convenient description of rooted non-separable planar maps. The involution $h$ on $\beta(1,0)$-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In this paper, we describe and enumerate fixed points of the involution $h$. Intriguingly, the fixed points are equinumerous with the fixed points under taking the dual map on rooted non-separable planar maps, even though the fixed points do not go to each other under the know (natural) bijection between the trees and the maps.