I'm currently reading a review article called *Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance* by *Jessica Striker*. In this article, Striker writes that there is a ''nice'' bijection between the set of noncrossing matchings of $2b$ points and $SYT(2 \times b)$, the set of standard Young tableaux of size $2 \times b$. Here are two examples of the bijection:

Unfortunately, I cannot seem to understand how exactly this bijection works.

As can be seen in the figure, given any pair $(i, j)$ with $i<j$ in the noncrossing matching, we want to map this to this to a tableau with $i$ on the first row and $j$ on the second row. But it is neither clear to me why this map should be injective nor surjective.

Catalan Numbers(Cambridge, 2015). Two-row standard Young tableaux are interpretation 168 (p44) and non crossing matchings (drawn in a line) are number 61 (p28). $\endgroup$ – Brian Hopkins May 30 '18 at 12:36