Fighting fish and two-stack sortable permutations

TL;DR

This paper establishes a bijection between fighting fish and two-stack sortable permutations, providing combinatorial insights and proving the algebraicity of their generating functions.

Contribution

It introduces a new recursive decomposition of two-stack sortable permutations and links it to fighting fish, extending known results on their generating functions.

Findings

Bijection between fighting fish and two-stack sortable permutations

Combinatorial explanations for previously known results

Proof of algebraicity of the generating function for two-stack sortable permutations

Abstract

In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed a new family of combinatorial objects called "fighting fish", which are counted by the same formula as more classical objects, such as two-stack sortable permutations and non-separable planar maps. In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations. With our bijection, we give combinatorial explanations of several results on fighting fish proved previously with generating functions. Using the decomposition of two-stack sortable permutations, we also prove the algebraicity of their generating function, extending a result of Bousquet-M\'elou (1998).