arXiv subjects

Anuran Maity

Publications and source records attributed to Anuran Maity.

6 recordsLinked to original sources

arXiv2026

On words that nearly $\theta$-commute

The Hamming distance between two equal length words $\alpha, \beta$ is the number of positions where $\alpha$ and $\beta$ differ. For $x, y \in \Sigma^*$ and antimorphic involution $\theta$, $x$ $\theta$-commutes with $y$, if the Hamming distance between $xy$ and $\theta(y)x$ is zero. When the Hamming distance between $xy$ and $\theta(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $\theta$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $\theta(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_\theta$ on $\Sigma^*$, where $x R_\theta y$ holds if and only if $x$ nearly $\theta$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $\theta$-commutes with $y$, and discuss various properties of this set.

math.CO↗

arXiv2026

Bounds on the closed-rich constant of infinite words

A finite word $w$ is called \textit{closed} if it has length at most 1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences in $w$. An infinite word $u$ is called \textit{closed-rich} if the infimum of all possible ratios between the number of closed factors within any factor $w$ of $u$ and square of the length of $w$ exists and is positive. We define this infimum as the closed-rich constant $C_u$ of the infinite closed-rich word $u$. Puzynina and Parshina (2024) proved that infinite closed-rich words exist. In this paper, we study possible values of closed-rich constants of infinite closed-rich words. In particular, we estimate the supremum $C_{sup}$ of the closed-rich constants of infinite closed-rich words: we show that $C_{sup} \leq 0.165952$. Besides that, we study the closed-rich constant $C_f$ of the Fibonacci word $f$ and show that $ 0.09519 \leq C_f\leq 0.10893 $. In particular, this gives a lower bound for $C_{sup}$: $ 0.09519 \leq C_{sup}$.

math.CO↗

arXiv2025

Mutually Abelian-Bordered Binary Words

A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and there exists a word that is a nonempty proper suffix of $u$ and prefix of $v$. Recently, Gabric studied the number of mutually bordered pairs. In this work, we extend the concept of mutually bordered pairs to abelian setting, and determine the number of mutually abelian-bordered pairs of binary words using lattice paths. We also find the number of unbordered pairs in this context.

math.CO↗

arXiv2023

Rich Words in the Block Reversal of a Word

The block reversal of a word $w$, denoted by $\mathtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words whose block reversal contains only rich words. We prove that for a binary word $w$, richness of all elements of $\mathtt{BR}(w)$ depends on $l(w)$, the length of the run sequence of $w$. We show that if all elements of $\mathtt{BR}(w)$ are rich, then $2\leq l(w)\leq 8$. We also provide the structure of such words.

math.CO↗

arXiv2022

Watson-Crick conjugates of words and languages

In this work, we explore the concept of Watson-Crick conjugates, also known as $\theta$-conjugates (where $\theta$ is an antimorphic involution), of words and languages. This concept extends the classical idea of conjugates by incorporating the Watson-Crick complementarity of DNA sequences. Our investigation initially focuses on the properties of $\theta$-conjugates of words. We then define $\theta$-conjugates of a language and study closure properties of certain families of languages under the $\theta$-conjugate operation. Furthermore, we analyze the iterated $\theta$-conjugate of both words and languages. Finally, we discuss the idea of $\theta$-conjugate-free languages and examine some decidability problems related to it.

cs.FL↗

arXiv2020

Theta palindromes in theta conjugates

A DNA string is a Watson-Crick (WK) palindrome when the complement of its reverse is equal to itself. The Watson-Crick mapping $θ$ is an involution that is also an antimorphism. $θ$-conjugates of a word is a generalisation of conjugates of a word that incorporates the notion of WK-involution $θ$. In this paper, we study the distribution of palindromes and Watson-Crick palindromes, also known as $θ$-palindromes among both the set of conjugates and $θ$-conjugates of a word $w$. We also consider some general properties of the set $C_θ(w)$, i.e., the set of $θ$-conjugates of a word $w$, and characterize words $w$ such that $|C_θ(w)|=|w|+1$, i.e., with the maximum number of elements in $C_θ(w)$. We also find the structure of words that have at least one (WK)-palindrome in $C_θ(w)$.

cs.FL↗