Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Enumeration of multipartite series-reduced trees

Medet Jumadildayev

TL;DR

The paper develops a generating-function framework for rooted multipartite labeled series-reduced trees, using Bell groups and compositional inverses to encode degree sequences and colorings. It establishes a central formula $P(m, t, x)=\left(t+\sum_{c=1}^m (x_c^{\langle -1 \rangle}(t)-t)\right)^{\langle -1 \rangle}$ that yields explicit enumerations, including connections to symbolic ultrametrics and increasingly labeled processes. It further provides closed-form and recursive expressions for both labeled and unlabeled variants, along with applications to mobiles and fully colored trees, and links to Riordan–Shannon refinements. The work reveals rich bijections between tree structures and combinatorial objects like chain-increasing binary trees and parallel process models, offering new counting formulas and insights for ultrametrics and process-graph analogies.

Abstract

We obtain a generating function for the degree sequences and colors of rooted multipartite labeled series-reduced trees. As an application of this result, we determine the number of symbolic ultrametrics (introduced by Böcker and Dress) and increasingly labeled processes. We also find that the number of multipartite labeled series-reduced trees and the colored chain-increasing binary trees are the same. We obtain the number of rooted multipartite unlabeled series-reduced trees. We also find a refinement of the result of Riordan and Shannon.

Enumeration of multipartite series-reduced trees

TL;DR

The paper develops a generating-function framework for rooted multipartite labeled series-reduced trees, using Bell groups and compositional inverses to encode degree sequences and colorings. It establishes a central formula that yields explicit enumerations, including connections to symbolic ultrametrics and increasingly labeled processes. It further provides closed-form and recursive expressions for both labeled and unlabeled variants, along with applications to mobiles and fully colored trees, and links to Riordan–Shannon refinements. The work reveals rich bijections between tree structures and combinatorial objects like chain-increasing binary trees and parallel process models, offering new counting formulas and insights for ultrametrics and process-graph analogies.

Abstract

We obtain a generating function for the degree sequences and colors of rooted multipartite labeled series-reduced trees. As an application of this result, we determine the number of symbolic ultrametrics (introduced by Böcker and Dress) and increasingly labeled processes. We also find that the number of multipartite labeled series-reduced trees and the colored chain-increasing binary trees are the same. We obtain the number of rooted multipartite unlabeled series-reduced trees. We also find a refinement of the result of Riordan and Shannon.

Paper Structure

This paper contains 9 sections, 17 theorems, 138 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Bell Groups
  3. Labeled Multipartite Series-reduced Trees
  4. Applications
  5. Enumeration of labeled multipartite trees and Symbolic Ultrametrics
  6. Multicolored chain-increasing binary trees
  7. Increasing Parallel Processes
  8. Unlabeled Multipartite Series-reduced Trees
  9. Acknowledgements

Key Result

Theorem 1.1

Let $m$ denote the number of colors. Additionally, for all colors $1 \leq c \leq m$ denote degree functions $x_c(t)$ and their compositional inverses $x_c^{\langle -1 \rangle}(t)$ as Then, the generating function for multipartite rooted labeled series-reduced trees on $m$ colors can be stated in terms of a compositional inverse of the sum of compositional inverses of degree functions as

Figures (6)

  • Figure 1: A tree with $9$ leaves and weight $\frac{t ^ 9}{9!}x_3 y_2 y_3 y_4.$
  • Figure 2: A labeled $3$-partite series-reduced tree with inner vertex weights.
  • Figure 3: Rooted bipartite labeled series-reduced trees with $3$ leaves.
  • Figure 4: Chain-increasing binary trees with $3$ chains.
  • Figure 5: An increasingly labeled tree with $5$ actions.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 29 more