Enumeration of symmetric arc diagrams

TL;DR

This paper develops recurrence relations for counting symmetric arc diagrams related to RNA structures, using combinatorial methods like Motzkin paths and bijections to ternary words, and analyzes their asymptotic behavior.

Contribution

It introduces new recurrence relations for symmetric arc diagrams associated with involutions and set partitions, connecting combinatorial structures to RNA secondary structures.

Findings

Derived recurrence relations for symmetric arc diagrams

Established bijections between noncrossing arc diagrams and ternary words

Analyzed asymptotic probabilities of symmetric structures with many nodes

Abstract

We give recurrence relations for the enumeration of symmetric elements within four classes of arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These arc diagrams are motivated by the study of RNA secondary structures. For example, classic RNA secondary structures correspond to 3412-avoiding involutions with no adjacent transpositions, and structures with base triples may be represented as partitions with crossings. Our results rely on combinatorial arguments. In particular, we use Motzkin paths to describe noncrossing arc diagrams that have no arc connecting two adjacent nodes, and we give an explicit bijection to ternary words whose length coincides with the sum of their digits. We also discuss the asymptotic behavior of some of the sequences considered here in order to quantify the extremely low probability of finding symmetric structures with a large number of nodes.