Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{d−3}$ are in bijection with the diagonals and sides of the $d$-gon, and the clusters are in bijection with triangulations of the $d$-gon, but I'm not really sure how to prove it and I don't think this is simply just the matter of counting.
I'd appreciate any ideas or references, or both.