Fighting fish

We introduce new combinatorial structures, called fighting fish, that generalize directed convex polyominoes by allowing them to branch out of the plane into independent substructures. On the one hand the combinatorial structure of fighting fish appears to be particularly rich: we show that their generating function with respect to the perimeter and number of tails is algebraic, and we conjecture a mysterious multivariate equidistribution property with the left ternary trees introduced by Del Lungo et al On the other hand, fighting fish provide a simple and natural model of random branching surfaces which displays original features: in particular, we show that the average area of a uniform random fighting fish with perimeter 2n is of order n 5/4: to the best of our knowledge this behaviour is non-standard and suggests that we have identified a new universality class of random structures.