  An endhered (end-adhered) pattern is a subset of arcs in matchings, such that
the corresponding starting points are consecutive and the same holds for the
ending points. Such patterns are in one-to-one correspondence with the
permutations. We focus on the occurrence frequency of such patterns in
matchings and real-world RNA structures with pseudoknots. We present
combinatorial results related to the distribution and asymptotic behavior of
the pattern 21, which corresponds to two consecutive stacked bonds frequently
encountered in RNA, and the pattern 12, representing the archetypal minimal
pseudoknot. We show that in matchings these two patterns are equidistributed,
which is quite different from what we can find in real-world RNAs. We also
examine the distribution of endhered patterns of size 3, showing how the
patterns change under the transformation called endhered twist. Finally, we
compute the distributions of endhered patterns of size 2 and 3 in real-world
secondary RNA structures with pseudoknots and discuss possible outcomes of our
study.
