Enumeration of Rooted Planar Triangulations with Respect to Diagonal Flips

We use the generating function approach to enumerate two families of rooted planar near-triangulations (2-connected, and 2-connected with no multiple edges) with respect to the number of flippable edges. It is shown that their generating functions are algebraic. Simple explicit expressions are obtained for the expected number of flippable edges in a random near-triangulation. Asymptotic estimates are obtained for the first two moments, which are then used to show that the numbers of flippable edges in a random near-triangulation and strict near-triangulation are sharply concentrated around 5n/2 and 9n/4, respectively.