A bijection between Tamari intervals and extended fighting fish

We introduce extended fighting fish as branching surfaces that can also be seen walks in the quarter plane defined by simple rewriting rules. The main result we present article is a direct bijection between and intervals of Tamari lattice exchanges multiple natural statistics. model includes recently introduced (Duchi et al., 2017) were shown to equinumerated with synchronized intervals. Using dual surface/walk points view on fish, show area statistic these corresponds distance (or maximal length chain) average uniform random size n, hence over set order n5/4, accordance earlier results for subclass ordinary fish.