TITLE: Non-nesting matchings and Catalan numbers
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: These are nonnesting partitions, which are usually defined as anti-chains in the root poset. Namely, the root poset (in $S_n$) is $\{ e_j-e_i : 1 \leq i < j \leq n \}$ with $e_k - e_j \leq e_{\ell} - e_i$ if $i \leq j < k \leq \ell$. Your arc diagrams turn into nonnesting partitions by $(i,j+1) \mapsto e_j-e_i$.
Googling "nonnesting partition" will give you a ton of hits.