On enumeration of tree-rooted planar cubic maps. II

TL;DR

This paper extends previous enumeration results of tree-rooted planar cubic maps by removing the requirement of a marked directed edge, providing more general formulas for counting such maps.

Contribution

It generalizes prior enumeration formulas to include maps without a marked directed edge, offering new combinatorial counting methods.

Findings

Derived new enumeration formulas for tree-rooted planar cubic maps without marked edges.

Confirmed that formulas are more complex but similar in structure to previous results.

Enhanced understanding of combinatorial structures in planar cubic maps.

Abstract

In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.