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Enumeration of symmetric arc diagrams

Juan B. Gil and Luis E. Lopez

Vol. 16 (2023), No. 1, 107–126

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.

symmetric arc diagrams, secondary structures, pattern-avoiding involutions, noncrossing partitions
Mathematical Subject Classification
Primary: 05A19
Received: 20 September 2021
Revised: 19 March 2022
Accepted: 23 March 2022
Published: 14 April 2023

Communicated by Arthur T. Benjamin
Juan B. Gil
Penn State Altoona
Altoona, PA
United States
Luis E. Lopez
Penn State Altoona
Altoona, PA
United States