The number of non-crossing matchings of sides of $2n$-gon (i.e. the number of ways to connect sides pairwise by non-intersecting paths) is $n$’th Catalan number, $\text{Cat}_n$.
How to prove combinatorially that $(n+2)\text{Cat}_{n+1}=(4n+2)\text{Cat}_n$ for this interpretation of Catalan numbers?
For example, proof of this recurrence for triangulations of $(n+2)$-gon is described in Wikipedia: after choosing a side of $(n+3)$-gon and contracting corresp. triangle we get an $(n+2)$-gon with one (marked and) oriented edge — I want something like this.
In principle, triangulations and non-crossing matchings are connected by a chain of bijections (e.g. matchings $\leftrightarrow$ balanced parentheses $\leftrightarrow$ binary trees $\leftrightarrow$ triangulations) and one can try to transfer the description from the previous paragraph via this chain. But this way quickly becomes too convoluted (for me, at least).