A bijection between Tamari intervals and extended fighting fish

Abstract

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