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Group Actions and Some Combinatorics on Words with $\mathbf{vtm}$

John Machacek

TL;DR

This paper extends combinatorics on words by using group actions to define orbit-based generalizations of repetition and factor complexity, introducing abelian powers and abelian complexity. It shows Sturmian words do not minimize these generalized complexities and uses the Walnut prover to prove new square-avoidance results for the vt m word, plus a precise relation between vt m complexity under a symmetric-group action and the abelian complexity of the pd word. The central finding is that the group-action factor complexity of vt m equals the abelian complexity of pd minus one, established by a morphic link and orbit counting; these results illuminate how symmetry actions shape repetition phenomena and connect to abelian complexity and known sequences in the OEIS.

Abstract

We introduce generalizations of powers and factor complexity via orbits of group actions. These generalizations include concepts like abelian powers and abelian complexity. It is shown that this notion of factor complexity cannot be used to recognize Sturmian words in general. Within our framework, we establish square avoidance results for the ternary squarefree Thue--Morse word $\mathbf{vtm}$. These results go beyond the usual squarefreeness of $\mathbf{vtm}$ and are proved using Walnut. Lastly, we establish a group action factor complexity formula for $\mathbf{vtm}$ that is expressed in terms of the abelian complexity of the period doubling word $\mathbf{pd}$.

Group Actions and Some Combinatorics on Words with $\mathbf{vtm}$

TL;DR

This paper extends combinatorics on words by using group actions to define orbit-based generalizations of repetition and factor complexity, introducing abelian powers and abelian complexity. It shows Sturmian words do not minimize these generalized complexities and uses the Walnut prover to prove new square-avoidance results for the vt m word, plus a precise relation between vt m complexity under a symmetric-group action and the abelian complexity of the pd word. The central finding is that the group-action factor complexity of vt m equals the abelian complexity of pd minus one, established by a morphic link and orbit counting; these results illuminate how symmetry actions shape repetition phenomena and connect to abelian complexity and known sequences in the OEIS.

Abstract

We introduce generalizations of powers and factor complexity via orbits of group actions. These generalizations include concepts like abelian powers and abelian complexity. It is shown that this notion of factor complexity cannot be used to recognize Sturmian words in general. Within our framework, we establish square avoidance results for the ternary squarefree Thue--Morse word . These results go beyond the usual squarefreeness of and are proved using Walnut. Lastly, we establish a group action factor complexity formula for that is expressed in terms of the abelian complexity of the period doubling word .

Paper Structure

This paper contains 7 sections, 15 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Some general observations
  4. Actions of subgroups
  5. Sturmian words
  6. Strengthening squarefreeness of $\mathbf{vtm}$
  7. Factor Complexity of $\mathbf{vtm}$ with Action of Symmetric Groups

Key Result

Proposition 3.1

Let $G' \leq G$ and $H'_{\ell} \leq H_{\ell}$ for some $\ell > 0$ be subgroups. If a word $\mathbf{w}$ avoids $(G, H_{\ell})$-$r$th powers, then $\mathbf{w}$ avoids $(G', H'_{\ell})$-$r$th powers

Figures (1)

  • Figure 1: An automaton accepting binary strings that represent lengths of certain $(S_3,1)$-squares in $\mathbf{vtm}$.

Theorems & Definitions (27)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • proof
  • Remark 3.7
  • ...and 17 more