Runs, Squares, Palindromes, and Unbordered Factors of a Family of Binary Pattern Sequences with the All-One Pattern

TL;DR

The paper studies the binary pattern sequences $s_m$ generated by the all-one pattern $P_m=1^m$ and analyzes maximal runs, squares, palindromes, and unbordered factors for $m\ge3$. It establishes that each $s_m$ is automatic via a $2m$-state DFAO and leverages a dictionary-style correspondence between binary strings and integers to connect automata structure with the $2$-kernel, the Jacobsthal sequence, and the Vile numbers, supplemented by computational explorations in $\text{Walnut}$ and $\text{Mathematica}$. Core contributions include a five-type classification of maximal run lengths, three lemmas characterizing square orders (with a conjecture that these cover all orders), and a study of the $n$-th largest maximal palindrome lengths, together with conjectures on unbordered factors and several open questions tied to higher arithmetic-hierarchy complexity. The approach blends automatic-proof techniques with exploratory pattern analysis, extending known results from Thue–Morse and Rudin–Shapiro to this broader family and illustrating the potential of a concatenation–addition viewpoint for proofs in combinatorics on words.

Abstract

This paper presents results on maximal runs, order of squares, palindromes, and unbordered factors of members of the family of binary pattern sequences with the all-one pattern. Restricting ourselves to binary pattern sequences with the all-one pattern with at least three ones, five categories of maximal run lengths and 3 categories of orders of squares are presented, palindromes with locally maximal length as well as palindromes with the second to fifth-largest palindrome lengths are described, and unbordered factors of lengths powers of two are presented. Interestingly, the characteristic functions of specified prefixes of sequences of the 2-kernel of these sequences can be formulated using the Vile and Jacobsthal sequences. Both Mathematica and Walnut are employed for exploratory pattern analysis. Proofs are based on a correspondence between binary strings under concatenation and integers under addition and multiplication. It is observed that this correspondence seems most efficacious for proofs of theorems whose statements are classified at low levels in the arithmetic hierarchy.

Figures (1)

• Figure 1: The DFAO for $s_4.$

Theorems & Definitions (30)

• Definition 1
• Lemma 2: Dictionary
• Theorem 3: Main
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof