A bijection between rooted planar maps and generalized fighting fish

TL;DR

This paper introduces generalized fighting fish, a new combinatorial model, and establishes a bijection with rooted planar maps via Mullin codes, linking two important classes of combinatorial objects.

Contribution

It extends fighting fish to a broader class and proves a bijection with rooted planar maps using Mullin codes, enriching combinatorial bijections.

Findings

Generalized fighting fish are exactly the Mullin codes of rooted planar maps.

Bijection between fighting fish and nonseparable rooted planar maps.

Connection to enumeration sequence /(n+1)(2n+1)   n

Abstract

The class of fighting fish is a recently introduced model of branching surfaces generalizing parallelogram polyominoes. We can alternatively see them as gluings of cells, walks on the square lattice confined to the quadrant or shuffle of Dyck words. With these different points of view, we introduce a natural extension of fighting fish that we call \emph{generalized fighting fish}. We show that generalized fighting fish are exactly the Mullin codes of rooted planar maps endowed with their unique rightmost depth-first search spanning tree, also known as Lehman-Lenormand code. In particular, this correspondence gives a bijection between fighting fish and nonseparable rooted planar maps, enriching the garden of bijections between classes of objects enumerated by the sequence $\frac{2}{(n+1)(2n+1)} \binom{3n}{n}$.