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Narad Rampersad

Publications and source records attributed to Narad Rampersad.

At least 19 recordsLinked to original sources
arXiv2026

Subword enumeration up to stack-sorting equivalence

Defant and Kravitz introduced generalizations of West's stack-sorting map $s$ from permutations to finite words. This raises questions as to how such generalizations could be applied in the field of combinatorics on words. The Defant-Kravitz generalizations of $s$ depend on how repeated occurrences of the same character within a word may be repositioned, according to their $\textsf{tortoise}$ and $\textsf{hare}$ operations. As demonstrated in this paper, these operations provide a natural way of extending abelian complexity functions for infinite sequences, in a way that gives light to structural properties associated with infinite words. We apply these new ideas to two famous infinite words: the paperfolding word and the Thue-Morse word. In the case of the Thue-Morse word, we discover an interesting connection to the previous work of several authors, such as de Luca and Varricchio, on the ``special'' factors of the Thue-Morse word. This may be seen as providing a basis for a new and interdisciplinary area linking the combinatorics about the stack-sorting of permutations with the field of combinatorics on words.

math.CO↗
arXiv2026

Complexity of Linear Subsequences of $k$-Automatic Sequences

We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences $(h(i))_{i \geq 0}$ and discuss their state complexity. We find a relationship between subword complexity of the interior sequence $(h'(i))_{i \geq 0}$ and state complexity of the linear subsequence $(h(ni+c))_{i \geq 0}$. We resolve a recent question of Zantema and Bosma about linear subsequences of $k$-automatic sequences with input in most-significant-digit-first format. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations or carrying out operations on automatic sequences.

cs.FL↗
arXiv2026

Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations.

cs.FL↗
arXiv2025

Low complexity binary words avoiding $(5/2)^+$-powers

Rote words are infinite words that contain $2n$ factors of length $n$ for every $n \geq 1$. Shallit and Shur, as well as Ollinger and Shallit, showed that there are Rote words that avoid $(5/2)^+$-powers and that this is best possible. In this note we give a structure theorem for the Rote words that avoid $(5/2)^+$-powers, confirming a conjecture of Ollinger and Shallit.

math.CO↗
arXiv2025

Reduced complexities for sequences over finite alphabets

Letting $w$ denote a finite, nonempty word, let $\text{red}(w)$ denote the word obtained from $w$ by replacing every subword $s$ of $w$ of the form $cc \cdots c$ for a given character $c$ (such that there is no character immediately to the left or right of $s$ equal to $c$) with $c$. Complexity functions for infinite words play important roles within combinatorics on words, and this leads us to introduce and investigate variants of the factor and abelian complexity functions using the given reduction operation. By enumerating words $v$ and $w$ of a given length $n \geq 0$ and associated with an infinite sequence over a finite alphabet such that $\text{red}(v)$ and $\text{red}(w)$ are equal or otherwise equivalent in some specified way, by analogy with the factor and abelian complexity functions, this may be seen as producing simplified versions of previously introduced complexity functions. We prove a recursion for the reduced factor complexity function $ρ_{\mathbf{t}}^{\text{red}}$ for the Thue-Morse sequence $\mathbf{t}$, giving us that $(ρ_{\mathbf{t}}^{\text{red}}(n) : n \in \mathbb{N})$ is a $2$-regular sequence, we prove an explicit evaluation for the reduced factor complexity function $ρ_{\mathbf{f}}^{\text{red}}$ for the (regular) paperfolding sequence $\mathbf{f}$, together with an evaluation for the reduced abelian complexity function $ρ_{\mathbf{f}}^{\text{ab}, \text{red}}$ for $\mathbf{f}$. We conclude with open problems concerning $ρ_{\mathbf{t}}^{\text{ab}, \text{red}}$.

math.CO↗
arXiv2024

Dyck Words, Pattern Avoidance, and Automatic Sequences

We study various aspects of Dyck words appearing in binary sequences, where $0$ is treated as a left parenthesis and $1$ as a right parenthesis. We show that binary words that are $7/3$-power-free have bounded nesting level, but this no longer holds for larger repetition exponents. We give an explicit characterization of the factors of the Thue-Morse word that are Dyck, and show how to count them. We also prove tight upper and lower bounds on $f(n)$, the number of Dyck factors of Thue-Morse of length $2n$.

cs.DM↗
arXiv2023

A small morphism giving Abelian repetition threshold less than 2

It is known that there are infinite words over finite alphabets with Abelian repetition threshold arbitrarily close to 1; however, the construction previously used involves huge alphabets. In this note we give a short cyclic morphism (length 13) over an 8-letter alphabet yielding an Abelian repetition threshold less than 1.8.

math.CO↗
arXiv2023

The analogue of overlap-freeness for the Fibonacci morphism

A $4^-$-power is a non-empty word of the form $XXXX^-$, where $X^-$ is obtained from $X$ by erasing the last letter. A binary word is called {\em faux-bonacci} if it contains no $4^-$-powers, and no factor 11. We show that faux-bonacci words bear the same relationship to the Fibonacci morphism that overlap-free words bear to the Thue-Morse morphism. We prove the analogue of Fife's Theorem for faux-bonacci words, and characterize the lexicographically least and greatest infinite faux-bonacci words.

math.CO↗
arXiv2023

Correlations of minimal forbidden factors of the Fibonacci word

If $u$ and $v$ are two words, the correlation of $u$ over $v$ is a binary word that encodes all possible overlaps between $u$ and $v$. This concept was introduced by Guibas and Odlyzko as a key element of their method for enumerating the number of words of length $n$ over a given alphabet that avoid a given set of forbidden factors. In this paper we characterize the pairwise correlations between the minimal forbidden factors of the infinite Fibonacci word.

math.CO↗
arXiv2023

Sums of products of binomial coefficients mod 2 and 2-regular sequences

Wu showed that certain sums of products of binomial coefficients modulo 2 are given by the run length transforms of several famous linear recurrence sequences, such as the positive integers, the Fibonacci numbers, the extended Lucas numbers, and Narayana's cows sequence. In this paper we show that the run length transform of such sequences are 2-regular sequences. This allows us to obtain Wu's results and some new ones using the computer program Walnut, eliminating the need for long technical proofs.

math.NT↗
arXiv2023

Antisquares and Critical Exponents

The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.

math.CO↗
arXiv2023

Rudin-Shapiro Sums Via Automata Theory and Logic

We show how to obtain, via a unified framework provided by logic and automata theory, many classical results of Brillhart and Morton on Rudin-Shapiro sums. The techniques also facilitate easy proofs for new results.

math.NT↗
arXiv2023

Complement Avoidance in Binary Words

The complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. We study infinite binary words $\bf w$ that avoid sufficiently large complementary factors; that is, if $x$ is a factor of $\bf w$ then $\overline{x}$ is not a factor of $\bf w$. In particular, we classify such words according to their critical exponents.

math.CO↗
arXiv2022

Properties of a Ternary Infinite Word

We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.

cs.DM↗
arXiv2021

The periodic complexity function of the Thue-Morse word, the Rudin-Shapiro word, and the period-doubling word

We revisit the periodic complexity function $h_{\bf w}(n)$ introduced by Mignosi and Restivo. This function gives the average of the first $n$ local periods of a recurrent infinite word ${\bf w}$. We give a different method than that of Mignosi and Restivo for computing the asymptotics of the periodic complexity function of the Thue-Morse word and show how to apply the method to other automatic sequences, like the Rudin-Shapiro word and the period-doubling word.

math.CO↗