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.
