There is a well-known bijection between (i) noncrossing partitions of the set {1,...,2n} where all blocks have size 2, and (ii) standard Young tableaux with n rows and 2 columns. There exist a Catalan number $C_n$ of them.
Define a noncrossing set of arcs between 2n points with $w_i$ arcs starting or ending at the point $i$. Loops are not allowed to occur. It seems to me that such patterns are in bijection with row-strict Young tableaux with $\frac12 \sum_{i=1}^{2n} w_i$ rows and 2 columns with weights $(w_1,...,w_{2n})$. One side of the bijection consists of listing all starting points on the left column and all ending points on the right column. Here is an example with weights (2,2,1,1,1,1):
I could not find any reference about it, perhaps I have the wrong keywords. I would be glad to have any reference talking about this kind of bijection.