  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.
