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Balanced Fibonacci word rectangles, and beyond

Jeffrey Shallit, Ingrid Vukusic

TL;DR

This work studies the balance of $m imes n$ word-rectangles built from key combinatorial words, starting with the Fibonacci word and extending to Sturmian words tied to quadratic irrationals, as well as Tribonacci and Thue–Morse words. It develops an automata-based framework, using Ostrowski and Zeckendorf representations and first-order logic, to decide which $(m,n)$ yield balanced blocks, and provides explicit automata (e.g., a 15-state Fibonacci bal automaton, 77-state Tribonacci automaton, and a 92-state Thue–Morse automaton) implemented in Walnut. The paper delivers both a precise automaton-based characterization for the Fibonacci case and readable Zeckendorf criteria, plus diversity results and parity-driven balance findings for the other words, illustrating a comprehensive, algorithmic approach to balance in word-rectangles across several well-studied morphic sequences. These methods offer practical tools for automatic verification and insight into the structure of balance phenomena in combinatorics on words, with potential extensions to further morphic words and numeration systems.

Abstract

Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational. Finally, we also examine the analogous question for the Tribonacci word and the Thue-Morse word.

Balanced Fibonacci word rectangles, and beyond

TL;DR

This work studies the balance of word-rectangles built from key combinatorial words, starting with the Fibonacci word and extending to Sturmian words tied to quadratic irrationals, as well as Tribonacci and Thue–Morse words. It develops an automata-based framework, using Ostrowski and Zeckendorf representations and first-order logic, to decide which yield balanced blocks, and provides explicit automata (e.g., a 15-state Fibonacci bal automaton, 77-state Tribonacci automaton, and a 92-state Thue–Morse automaton) implemented in Walnut. The paper delivers both a precise automaton-based characterization for the Fibonacci case and readable Zeckendorf criteria, plus diversity results and parity-driven balance findings for the other words, illustrating a comprehensive, algorithmic approach to balance in word-rectangles across several well-studied morphic sequences. These methods offer practical tools for automatic verification and insight into the structure of balance phenomena in combinatorics on words, with potential extensions to further morphic words and numeration systems.

Abstract

Following a recent paper of Anselmo et al., we consider rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational. Finally, we also examine the analogous question for the Tribonacci word and the Thue-Morse word.

Paper Structure

This paper contains 8 sections, 13 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sturmian words
  3. Finite automata
  4. Another version of the criterion for balance
  5. Diverse rectangles
  6. Balanced rectangles more generally
  7. The Thue--Morse word
  8. Walnut commands

Key Result

Lemma 1

If $(a_i)_{i \geq 0}$ is a Sturmian word corresponding to the irrational number $\alpha$, then in the matrix ${\cal A}_\alpha = (a_{i,j})_{i,j\geq 0}$ the $m \times n$ blocks are balanced if and only if the sequence $\Delta(i,m,n)$ has no blocks of the form $1,0,0,\ldots, 0,1$ or $-1,0,0,\ldots,0, -

Figures (2)

  • Figure 1: Automaton accepting balanced $(m,n)$ pairs for the Fibonacci word.
  • Figure 2: Automaton accepting balanced $(m,n)$ pairs for the Fibonacci word with $m \leq n$.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 14 more