[Paper Review] An involution on bicubic maps and β(0,1)-trees

Anders Claesson, Sergey Kitaev+1 more|arXiv (Cornell University)|Oct 11, 2012

Graph theory and applications7 references3 citations

TL;DR

This paper introduces an involution on β(0,1)-trees—combinatorial structures in bijection with bicubic maps—that proves the joint equidistribution of the statistics f1r3 and f3r2 on bicubic maps. The involution swaps root and rmod statistics on β(0,1)-trees, establishing symmetry in bivariate generating functions. The key result is that the joint distribution of (f1r3, f3r2) is invariant under exchange of the two statistics, extending earlier equidistribution results to a stronger symmetry.

ABSTRACT

Bicubic maps are in bijection with β(0,1)-trees. We introduce two new ways of decomposing β(0,1)-trees. Using this we define an endofunction on β(0,1)-trees, and thus also on bicubic maps. We show that this endofunction is in fact an involution. As a consequence we are able to prove some surprising results regarding the joint equidistribution of certain pairs of statistics on trees and maps. Finally, we conjecture the number of fixed points of the involution.

Motivation & Objective

• To establish a stronger equidistribution result for statistics on bicubic maps beyond individual equidistribution.
• To define a new involution on β(0,1)-trees that swaps key statistics and preserves the tree structure.
• To prove that the joint distribution of (f1r3, f3r2) on bicubic maps is symmetric under exchange of the two statistics.
• To conjecture the number of fixed points of the involution, linking it to Catalan numbers and distinguished nodes in β(0,1)-trees.
• To extend prior work on β(1,0)-trees and non-separable maps to the β(0,1)-tree and bicubic map setting.

Proposed method

• Define β(0,1)-trees as rooted plane trees with integer labels satisfying specific sum constraints: leaves are 0, root label is one more than sum of children, and internal nodes exceed the sum by at most one.
• Introduce two new decompositions of β(0,1)-trees: via the root label and via the number of children of the root (sub(T)).
• Define an endofunction g on β(0,1)-trees that swaps the root label and the rmod statistic (residue modulo 1 of the sum of children's labels), and prove it is an involution.
• Use the known bijection between β(0,1)-trees and bicubic maps to translate the involution to the map setting and prove symmetry of bivariate generating functions.
• Provide two proofs: one via generating functions and one combinatorial, using the involution g to show joint equidistribution.
• Conjecture that the number of fixed points of g is 2^{n-1} times the Catalan number C_{floor(n/2)}, linking it to trees with a distinguished excessive node.

Experimental results

Research questions

• RQ1 Is the joint distribution of the statistics f1r3 and f3r2 on bicubic maps symmetric under exchange of the two statistics?
• RQ2 Can an involution on β(0,1)-trees be constructed that swaps the root label and the rmod statistic while preserving the tree structure?
• RQ3 What is the number of fixed points of this involution, and can it be expressed in closed form?
• RQ4 Does the symmetry of the joint distribution extend to other pairs of statistics on β(0,1)-trees?
• RQ5 Is there a bijection between fixed points of the involution and β(0,1)-trees with a distinguished excessive node?

Key findings

• The involution g on β(0,1)-trees is proven to be an involution, meaning g(g(T)) = T for all trees T.
• The involution g swaps the root label and the rmod statistic, and satisfies root(g(T)) = rmod(T) and rmod(g(T)) = root(T).
• As a consequence, the joint distribution of (root, rmod) on β(0,1)-trees is symmetric: ∑ x^{root(T)} y^{rmod(T)} = ∑ x^{rmod(T)} y^{root(T)} over all trees with n nodes.
• This symmetry implies the joint equidistribution of (f1r3, f3r2) and (f3r2, f1r3) on bicubic maps, proving Theorem 2.
• The number of fixed points of the involution g is conjectured to be 2^{n-1} C_{floor(n/2)}, where C_k is the k-th Catalan number.
• The conjecture is supported by computation for n ≤ 12 and is equivalent to a bijection between fixed points and β(0,1)-trees with a distinguished excessive node on floor(n/2)+1 nodes.