A circular order on edge-coloured trees and RNA m-diagrams

TL;DR

This paper establishes a bijection between edge-coloured trees with a fixed circular order and RNA m-diagrams, providing enumeration and structural insights into these combinatorial objects related to RNA secondary structures.

Contribution

It introduces a novel bijection between labelled m-edge-coloured trees with a fixed circular order and RNA m-diagrams, and characterizes these trees as equivalence classes under a specific operation.

Findings

Bijection between trees and RNA diagrams for fixed order

Enumeration formulas for these combinatorial objects

Characterization of trees as equivalence classes under an operation

Abstract

We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as an induction in the context of interval exchange transformations.