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Computing the k-binomial complexity of generalized Thue--Morse words

M. Golafshan, M. Rigo, M. Whiteland

TL;DR

This work addresses the k-binomial complexities of generalized TM words by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word and characterizing k-binomial equivalence among factors of generalized TM words.

Abstract

Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$ to the number of k-binomial equivalence classes represented by its factors of length n. The Thue--Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for $k\geq 3$ remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period $m^k$. We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.

Computing the k-binomial complexity of generalized Thue--Morse words

TL;DR

This work addresses the k-binomial complexities of generalized TM words by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word and characterizing k-binomial equivalence among factors of generalized TM words.

Abstract

Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer to the number of k-binomial equivalence classes represented by its factors of length n. The Thue--Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period . We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.

Paper Structure

This paper contains 16 sections, 42 theorems, 140 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Previously known results on generalized Thue--Morse words
  3. Main results
  4. Key Points of Our Proof Strategy
  5. Compilation of Preliminary Results
  6. Ability to Discern k-Binomially Non-Equivalent Factors
  7. Recognizability and Structure of Factors
  8. Counting Classes of a New Equivalence Relation
  9. Characterizing Binomial Equivalence in tm
  10. The base case
  11. The induction step
  12. Abelian Complexity for Short Factors
  13. Description of the Abelian Rauzy Graphs
  14. When ell < m
  15. When ell >= m
  16. ...and 1 more sections

Key Result

proposition 1

Let $\mathbf{w}$ denote a fixed point of a Parikh-collinear morphism. For any $k \geqslant 1$, there exists a constant $C_k \in \mathop{\mathrm{\mathbb{N}}}\nolimits$ satisfying $\mathsf{b}_{\mathbf{w}}^{(k)} (n)\leqslant C_{k}$ for all $n \in \mathop{\mathrm{\mathbb{N}}}\nolimits$.

Figures (7)

  • Figure 1: The first few values of the factor complexity (dashed), $2$-, and $3$-binomial complexities of $\mathbf{t}_3$.
  • Figure 2: Illustrating the situation $|u|=|v|$ and $s_{_{_U}}$ or $s_{_{V}}$ non-empty.
  • Figure 3: Illustrating the situation $|u|+1=|v|$.
  • Figure 6: Abelian Rauzy graph $G_{6,4}$ of order $4$ for $\mathbf{t}_6$.
  • Figure 7: Abelian Rauzy graph $G_{5,4}$ of order $4$ for $\mathbf{t}_5$.
  • ...and 2 more figures

Theorems & Definitions (79)

  • definition 1
  • definition 2: RigoSalimov
  • definition 3
  • definition 4
  • proposition 1: RSW
  • theorem 1.1
  • theorem 1.2: ChenWen2019
  • theorem 1.3: LLR
  • theorem 1.4: ChenWen2024
  • theorem 1.5
  • ...and 69 more