[Paper Review] Bijective counting of tree-rooted maps and shuffles of parenthesis systems

Olivier Bernardi|arXiv (Cornell University)|Jan 27, 2006

Stochastic processes and statistical mechanics5 references79 citations

TL;DR

This paper presents a direct, non-recursive bijection between tree-rooted maps of size $n$ and pairs consisting of a binary tree and a non-crossing partition, explaining the long-standing enumerative formula $C_n C_{n+1}$, where $C_n$ is the $n$th Catalan number. The bijection is shown to be isomorphic to a recursive shuffle construction of parenthesis systems by Cori, Dulucq, and Viennot, thereby providing a natural, geometric interpretation of their algebraic result in terms of planar maps and lattice walks.

ABSTRACT

The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.

Motivation & Objective

• To provide a direct, non-recursive bijection between tree-rooted maps and pairs of a tree and a non-crossing partition, offering a geometric explanation for the formula $C_n C_{n+1}$.
• To resolve Mullin's long-standing request for a bijective explanation of the enumeration of tree-rooted maps.
• To establish isomorphism between the new bijection and the recursive shuffle construction of Cori, Dulucq, and Viennot on parenthesis systems.
• To clarify the connection between tree-rooted maps, shuffles of parenthesis systems, and quarter-plane lattice walks.

Proposed method

• Define a bijection between tree-rooted maps of size $n$ and pairs of a tree (size $n$) and a non-crossing partition (size $n+1$), using a tour-based encoding of the spanning tree.
• Construct a recursive mapping from prefix-shuffles of parenthesis systems to partition-trees via active/inactive vertices and edge insertions.
• Use a binary tree encoding $\lambda_1$ to map shuffles to binary trees, then apply a transformation $\theta$ to obtain the final tree structure.
• Prove that the resulting partition-tree matches the tree from the bijection via induction on prefix-shuffles, maintaining active/inactive vertex correspondence.
• Show that the vertex order and parent-child relations in the partition-tree match those in the $\theta \circ \lambda_1$ tree, establishing isomorphism.
• Use the correspondence between parenthesis shuffles and quarter-plane walks to interpret the result as a counting mechanism for such walks.

Experimental results

Research questions

• RQ1 Can a direct, non-recursive bijection be constructed between tree-rooted maps and pairs of trees of size $n$ and $n+1$?
• RQ2 Is the known recursive shuffle construction of Cori, Dulucq, and Viennot isomorphic to a geometric, map-based bijection?
• RQ3 How can the enumeration formula $C_n C_{n+1}$ for tree-rooted maps be naturally explained via planar maps and non-crossing structures?
• RQ4 What is the precise correspondence between the recursive shuffle construction and the geometric structure of tree-rooted maps?

Key findings

• A direct, non-recursive bijection is constructed between tree-rooted maps of size $n$ and pairs of a tree (size $n$) and a non-crossing partition (size $n+1$), explaining the formula $C_n C_{n+1}$.
• The bijection is proven to be isomorphic to the recursive shuffle construction of Cori, Dulucq, and Viennot via the encoding of tree-rooted maps as shuffles of two parenthesis systems.
• The construction maintains vertex activity, order, and parent-child structure across the bijection, ensuring consistency with the recursive tree growth process.
• The method provides a geometric interpretation of the shuffle construction, making the combinatorial result more intuitive and natural in the context of planar maps.
• The correspondence between tree-rooted maps and quarter-plane walks is explicitly established, with the bijection offering a new way to count such walks.
• The proof relies on an inductive construction on prefix-shuffles, showing that the partition-tree structure matches the $\theta \circ \lambda_1$ tree at every step.