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

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.

💡 Research Summary

The paper introduces a novel combinatorial invariant – the circular order – for labelled trees whose edges are coloured with m distinct colours and which contain exactly k vertices. Each vertex is incident to precisely one edge of each colour, so the tree is an m‑regular, m‑coloured tree. By embedding the vertices on a circle in the natural numerical order (1,2,…,k) and tracing around the circle following edges of a fixed colour i, one obtains a cyclic permutation π_i. The collection {π_1,…,π_m} is called the circular order profile of the tree.

The authors then define RNA m‑diagrams, a family of planar diagrams that generalise classical RNA secondary structures. An RNA m‑diagram of degree k places k points equally spaced on a circle and draws non‑crossing arcs of colour i (1 ≤ i ≤ m) connecting pairs of points. Arcs of the same colour never intersect, while arcs of different colours may be interleaved. Such diagrams can be interpreted as secondary structures in which m different types of base‑pairing or interaction are allowed.

The central result is a bijection between the set 𝒯_{k,m}(σ) of m‑coloured trees on k vertices that realise a fixed circular order σ (and therefore a fixed profile) and the set 𝒟_{k,m}(σ) of RNA m‑diagrams of degree k with the same circular order. The bijection is constructed explicitly: each vertex of the tree corresponds to a point on the circle, and each edge of colour i corresponds to an arc of colour i. The acyclicity of the tree guarantees that the arcs are non‑crossing, while the non‑crossing condition of the diagram forces the underlying graph to be a tree. The authors verify that the mapping preserves the circular order in both directions, establishing a one‑to‑one correspondence.

Having identified the bijection, the paper proceeds to enumerate the objects. Using generating‑function techniques and Lagrange inversion, the authors show that the number of m‑coloured trees (or equivalently RNA m‑diagrams) with a prescribed circular order is

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