It is well-known that the number of non-nesting perfect matchings on $2n$ points is given by the Catalan number $C_n$; see part (a) of the figure below. This is item e^5 in Stanley's list (http://www-math.mit.edu/~rstan/ec/catadd.pdf).
[The following section has been edited to account for an initial mistake in the description.] Now I am interested in non-nesting arc diagrams on $n+1$ points, where no arc connects two neighboring points (two arcs may meet in the same point, one arc from above and one arc from below, so the arcs may not form a matching); see part (b) of the figure below. These diagrams are also counted by $C_n$, and it is easy to prove this.
This must be a known fact, but the second type of arc diagrams is not in Stanley's book (maybe I overlooked it), so who has references for this type of arc diagrams with regards to Catalan numbers?