Enumeration of symmetric arc diagrams

Juan B. Gil, Luis E. Lopez

Research output: Contribution to journalArticlepeer-review


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 be-havior 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.

Original languageEnglish (US)
Pages (from-to)107-126
Number of pages20
Issue number1
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

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