A bijection between Tamari intervals and extended fighting fish

TL;DR

This paper establishes a bijection between extended fighting fish and Tamari lattice intervals, linking combinatorial structures with lattice theory and revealing new statistical correspondences and average size behaviors.

Contribution

It introduces extended fighting fish and demonstrates a direct bijection with Tamari intervals, connecting surface and walk models with lattice statistics.

Findings

Bijection exchanges natural statistics between models.

Area statistics correspond to Tamari interval distances.

Average size of extended fighting fish scales as n^{5/4}.

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, Guerrini, Rinaldi, Schaeffer 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 statistics on these fish corresponds to the distance statistics (or maximal length of a chain) in Tamari invervals. 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 $n^{5/4}$, in accordance with earlier result for the subclass fighting fish.