Enumeration of weighted plane trees by a permutation model

TL;DR

This paper provides a combinatorial proof for counting weighted bi-colored plane trees with labeled vertices, using a permutation-based geometric construction, and explores algebraic relationships involving Stirling numbers.

Contribution

It offers a purely combinatorial, constructive bijection proof for Kochetkov's enumeration formula, applicable to real-valued weights, and connects the problem to partition orders and Stirling numbers.

Findings

Bijection proof of the enumeration formula

Constructive geometric encoding of trees as permutations

Algebraic relationships with Stirling numbers

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.