Enumeration of weighted plane trees by a permutation model

TL;DR

The paper provides a constructive, bijective proof of Kochetkov's enumeration formula for labeled weighted bi-colored plane trees with full passports, valid for real edge weights. Central to the approach is a combing map that encodes trees as permutations and a folding map that recovers permutations from twice-marked trees, yielding a bijection between tree structures and a distinguished subset of permutations (tree permutations). By separating the decomposable and non-decomposable passport cases and introducing positive permutations, the authors derive explicit counting formulas that involve partitions and the Stirling numbers of the second kind, ultimately showing |Tree(Ξ_F)| equals a partition-sum expression and, in the non-decomposable case, reduces to (|Ξ_F|−2)!. The method is fully combinatorial and constructive, extending naturally to real edge weights and providing an algorithmic path to enumerate all LWBP-trees of a prescribed full passport. The work also connects the enumeration problem to nuanced algebraic structures on partitions and has potential implications for related meromorphic-differential frameworks on the Riemann sphere.

Abstract

This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and totally constructive, remaining valid for real-valued edge weights. A central process is a geometric construction that directly encodes each tree as a permutation. We also exhibit algebraic relationships between the enumeration problem, the partial order on partitions of vertices and the Stirling numbers of the second kind. Some computation examples are presented as appendices.

Figures (6)

• Figure 1: Some LWBP-trees of various kinds of passports, all with three vertices of weight $+1$ and one vertex of weight $-3$.
• Figure 2: The combing map.
• Figure 3: Three consecutive edges on the closed path.
• Figure 4: The folding map.
• Figure 5: Decomposition of a positive permutation.

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Definition 1.6
• proof : Proof of Theorem \ref{['thm:passport_simple']}
• Remark 1.7
• Definition 1.8
• Proposition 1.9