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Prime Factorization of Meanders

Y. Belousov

TL;DR

The paper addresses counting open meanders by introducing a prime-factorization into two prime classes—iterated snakes and irreducible meanders—within a 2-colored operad framework. It shows that every open meander admits a canonical rooted-tree construction using these two building blocks, with insertions governed by operadic composition, and derives a generating-function equation that expresses all meanders in terms of the irreducible and iterated-snakes components. The authors provide an explicit generating function for iterated snakes, establish efficient enumeration for this class, and present partial, computational results for irreducible meanders, along with asymptotic analyses that separate growth rates by class. These results create a structural and computational framework for meander enumeration and reveal connections to polyomino enumeration and related combinatorial problems, offering new avenues for asymptotic study in this long-standing counting problem.

Abstract

In this paper, we introduce a prime factorization of open meanders, articulated through the framework of 2-colored operads. We demonstrate that each open meander can be canonically constructed from building blocks of two types: iterated snakes and irreducible meanders. We find out that iterated snakes allow efficient enumeration, and thus the problem of enumerating meanders reduces to the problem of enumerating irreducible meanders. Additionally, we present some results concerning the asymptotic of meanders of both classes.

Prime Factorization of Meanders

TL;DR

The paper addresses counting open meanders by introducing a prime-factorization into two prime classes—iterated snakes and irreducible meanders—within a 2-colored operad framework. It shows that every open meander admits a canonical rooted-tree construction using these two building blocks, with insertions governed by operadic composition, and derives a generating-function equation that expresses all meanders in terms of the irreducible and iterated-snakes components. The authors provide an explicit generating function for iterated snakes, establish efficient enumeration for this class, and present partial, computational results for irreducible meanders, along with asymptotic analyses that separate growth rates by class. These results create a structural and computational framework for meander enumeration and reveal connections to polyomino enumeration and related combinatorial problems, offering new avenues for asymptotic study in this long-standing counting problem.

Abstract

In this paper, we introduce a prime factorization of open meanders, articulated through the framework of 2-colored operads. We demonstrate that each open meander can be canonically constructed from building blocks of two types: iterated snakes and irreducible meanders. We find out that iterated snakes allow efficient enumeration, and thus the problem of enumerating meanders reduces to the problem of enumerating irreducible meanders. Additionally, we present some results concerning the asymptotic of meanders of both classes.
Paper Structure (14 sections, 22 theorems, 46 equations, 15 figures, 1 table)

This paper contains 14 sections, 22 theorems, 46 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Meander permutation
  4. Submeanders, and the insertion of meanders
  5. The structure of a 2-colored operad
  6. Decomposition
  7. Preliminary lemmas
  8. Description of the factorization
  9. Equation for the generating function.
  10. Iterated snakes
  11. Number sequences associated with iterated snakes
  12. Irreducible meanders
  13. Asymptotics of the numbers of irreducible meanders
  14. Table with meander numbers

Key Result

Theorem 1

The set $\mathfrak{M} = \bigcup\limits_{n\geq 0, k \geq 0} \mathfrak{M}_{n,k}$ together with the set of operations form a 2-colored operad.

Figures (15)

  • Figure 1: Examples of meanders.
  • Figure 2: Meanders that would be equivalent if boundary points are not part of the data.
  • Figure 3: Examples of singular meanders.
  • Figure 4: Examples of submeanders.
  • Figure 5: Examples of meanders and the Hasse diagrams of their posets of submeanders.
  • ...and 10 more figures

Theorems & Definitions (66)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Remark 4
  • Definition 3
  • Remark 5
  • Remark 6
  • Definition 4
  • ...and 56 more