arXiv subjects

Josef Rukavicka

Publications and source records attributed to Josef Rukavicka.

At least 19 recordsLinked to original sources

arXiv2025

Subexponential upper bound on the number of rich words

Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that $\lim_{n\rightarrow\infty} \sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to now, no subexponential upper bound on $R(n)$ has been presented. The current paper fills this gap. Let $\frac{1}{2}<\lambda<1$ and $\gamma>1$ be real constants, let $q$ be the size of the alphabet, and let $\phi$ be a positive function with $\lim_{n\rightarrow\infty}\phi(n)=\infty$ and $\lim_{n\rightarrow\infty}\frac{n}{\phi(n)}=\infty$. Let $\ln^*(x)$ denote the iterated logarithm of $x>0$. We prove that there are $n_0$ and $c>0$ such that if $n>n_0$, \[f(n)=\sqrt[\gamma]{c\ln^*{(\frac{n}{\phi(n)}}\ln{q})}\quad\mbox{ and }\quad B(n)=q^{\frac{n}{\phi(n)}+\frac{n}{(2\lambda)^{f(n)-1}}}\mbox{}\] then $\lim_{n\rightarrow\infty}\sqrt[n]{B(n)}=1$ and $R(n)\leq B(n)$.

math.CO↗

arXiv2024

Palindromic length of infinite aperiodic words

The palindromic length of the finite word $v$ is equal to the minimal number of palindromes whose concatenation is equal to $v$. It was conjectured in 2013 that for every infinite aperiodic word $x$, the palindromic length of its factors is not bounded. We prove this conjecture to be true.

math.CO↗

arXiv2023

Restivo Salemi property for $\alpha$-power free languages with $\alpha\geq 5$ and $k\geq 3$ letters

In 2009, Shur published the following conjecture: Let $L$ be a power-free language and let $e(L)\subseteq L$ be the set of words of $L$ that can be extended to a bi-infinite word respecting the given power-freeness. If $u, v \in e(L)$ then $uwv \in e(L)$ for some word $w$. Let $L_{k,\alpha}$ denote an $\alpha$-power free language over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove the conjecture for the languages $L_{k,\alpha}$, where $\alpha\geq 5$ and $k\geq 3$.

cs.FL↗

arXiv2023

Note on dissecting power of regular languages

Let $c>1$ be a real constant. We say that a language $L$ is $c$-\emph{constantly growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is $c$-\emph{geometrically growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given a language $L$, we say that $L$ is $REG$-\emph{dissectible} if there is a regular language $R$ such that $\vert L\setminus R\vert=\infty$ and $\vert L\cap R\vert=\infty$. In 2013, it was shown that every $c$-constantly growing language $L$ is $REG$-dissectible. In 2023, the following open question has been presented: "Is the family of geometrically growing languages $REG$-dissectible?" We construct a $c$-geometrically growing language $L$ that is not $REG$-dissectible. Hence we answer negatively to the open question.

cs.FL↗

arXiv2023

Dissecting power of intersection of two context-free languages

We say that a language $L$ is \emph{constantly growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is \emph{geometrically growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given two infinite languages $L_1,L_2$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_2\setminus L_1\vert=\infty$ and $\vert L_1\cap L_2\vert=\infty$. In 2013, it was shown that for every constantly growing language $L$ there is a regular language $R$ such that $R$ dissects $L$. In the current article we show how to dissect a geometrically growing language by a homomorphic image of intersection of two context-free languages. Consider three alphabets $Γ$, $Σ$, and $Θ$ such that $\vert Σ\vert=1$ and $\vert Θ\vert=4$. We prove that there are context-free languages $M_1,M_2\subseteq Θ^*$, an erasing alphabetical homomorphism $π:Θ^*\rightarrow Σ^*$, and a nonerasing alphabetical homomorphism $φ: Γ^*\rightarrow Σ^*$ such that: If $L\subseteq Γ^*$ is a geometrically growing language then there is a regular language $R\subseteq Θ^*$ such that $φ^{-1}\left(π\left(R\cap M_1\cap M_2\right)\right)$ dissects the language $L$.

cs.FL↗

arXiv2022

Property of upper bounds on the number of rich words

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. Let $q\geq 2$ be the size of the alphabet. Let $R(n)$ be the number of rich words of length $n$. Let $d>1$ be a real constant and let $ϕ, ψ$ be real functions such that \begin{itemize}\item there is $n_0$ such that $2ψ(2^{-1}ϕ(n))\geq dψ(n)$ for all $n>n_0$, \item $\frac{n}{ϕ(n)}$ is an upper bound on the palindromic length of rich words of length $n$, and \item $\frac{x}{ψ(x)}+\frac{x\ln{(ϕ(x))}}{ϕ(x)}$ is a strictly increasing concave function. \end{itemize} We show that if $c_1,c_2$ are real constants and $R(n)\leq q^{c_1\frac{n}{ψ(n)}+c_2\frac{n\ln(ϕ(n))}{ϕ(n)}}$ then for every real constant $c_3>0$ there is a positive integer $n_0$ such that for all $n>n_0$ we have that \[R(n)\leq q^{(c_1+c_3)\frac{n}{dψ(n)}+c_2\frac{n\ln(ϕ(n))}{ϕ(n)}(1+\frac{1}{c_2\ln{q}}+c_3)}\mbox{.}\]

math.CO↗

arXiv2022

Upper bound for the number of privileged words

A non-empty word $w$ is a \emph{border} of a word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ is \emph{privileged} if $\vert u\vert\leq 1$ or if $u$ has a privileged border $w$ that appears exactly twice in $u$. Peltomäki (2016) presented the following open problem: ``Give a nontrivial upper bound for $B(n)$'', where $B(n)$ denotes the number of privileged words of length $n$. Let $\ln^{[0]}{(n)}=n$ and let $\ln^{[j]}{(n)}=\ln{(\ln^{[j-1]}{(n)})}$, where $j,n$ are positive integers. We show that if $q>1$ is a size of the alphabet and $j\geq 3$ is an integer then there are constants $α_j$ and $n_j$ such that \[B(n)\leq α_j\frac{q^{n}\sqrt{\ln{n}}}{\sqrt{n}}\ln^{[j]}{(n)}\prod_{i=2}^{j-1}\sqrt{\ln^{[i]}(n)}\mbox{, where }n\geq n_j\mbox{.}\] This result improves the upper bound of Rukavicka (2020).

math.CO↗

arXiv2022

Construction of a bi-infinite power free word with a given factor and a non-recurrent letter

Let $L_{k,α}^{\mathbb{Z}}$ denote the set of all bi-infinite $α$-power free words over an alphabet with $k$ letters, where $α$ is a positive rational number and $k$ is positive integer. We prove that if $α\geq 5$, $k\geq 3$, $v\in L_{k,α}^{\mathbb{Z}}$, and $w$ is a finite factor of $v$, then there are $\widetilde v\in L_{k,α}^{\mathbb{Z}}$ and a letter $x$ such that $w$ is a factor of $\widetilde v$ and $x$ has only a finitely many occurrences in $\widetilde v$.

cs.FL↗

arXiv2021

Palindromic factorization of rich words

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w_1, w_2,\dots,w_p$ such that $w=w_pw_{p-1}\cdots w_1$ and $w_i$ is the longest palindromic suffix of $w_pw_{p-1}\cdots w_i$, where $1\leq i\leq p$. This palindromic factorization is called \emph{UPS-factorization}. Let $luf(w)=p$ be \emph{the length of UPS-factorization} of $w$. In 2017, it was proved that there is a constant $c$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then $luf(w)\leq c\frac{n}{\ln{n}}$. We improve this result as follows: There are constants $μ, π$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then \[luf(w)\leq μ\frac{n}{e^{π\sqrt{\ln{n}}}}\mbox{.}\] The constants $c,μ,π$ depend on the size of the alphabet.

math.CO↗

arXiv2021

Secretary problem and two almost the same consecutive applicants

We present a new variant of the secretary problem. Let $A$ be a totally ordered set of $n$ \emph{applicants}. Given $P\subseteq A$ and $x\in A$, let $rr(P,x)=\vert\{z\in P \mid z\leq x\}\vert\mbox{ }$ be the \emph{relative rank of} $x$ \emph{with regard to} $P$, and let $rr_n(x)=rr(A,x)$. Let $x_1,x_2,\dots,x_n\in A$ be a random sequence of distinct applicants. The aim is to select $1<j\leq n$ such that $rr_n(x_{j-1})-rr_n(x_j)\in\{-1,1\}$. Let $α$ be a real constant with $0<α<1$. Suppose the following stopping rule $τ_n(α)$: reject first $αn$ applicants and then select the first $x_j$ such that $rr(P_j,x_{j-1})-rr(P_j,x_j)\in\{-1,1\}$, where $P_j=\{x_i\mid 1\leq i\leq j\}$. Let $p_{n,τ}(α)$ be the probability that $τ_n(α)$ selects $x_j$ such that $rr_n(x_{j-1})-rr_n(x_j)\in\{-1,1\}$. We show that \[\lim_{n\rightarrow\infty}p_{n,τ}(α)\leq \lim_{n\rightarrow\infty}p_{n,τ}\left(\frac{1}{2}\right)=\frac{1}{2}\mbox{.}\]

math.PR↗

arXiv2021

Palindromic Length and Reduction of Powers

Given a nonempty finite word $v$, let $PL(v)$ be the palindromic length of $v$; it means the minimal number of palindromes whose concatenation is equal to $v$. Let $v^R$ denote the reversal of $v$. Given a finite or infinite word $y$, let $Fac(y)$ denote the set of all finite factors of $y$ and let $maxPL(y)=\max\{PL(t)\mid t\in Fac(y)\}$. Let $x$ be an infinite non-ultimately periodic word with $maxPL(x)=k<\infty$ and let $u\in Fac(x)$ be a primitive nonempty factor such that $u^5$ is recurrent in $x$. Let $Ψ(x,u)=\{t\in Fac(x)\mid u,u^R\not\in Fac(t)\}\mbox{.}$ We construct an infinite non-ultimately periodic word $\overline x$ such that $u^5, (u^R)^5\not\in Fac(\overline x)$, $Ψ(x,u)\subseteq Fac(\overline x)$, and $maxPL(\overline x)\leq 3k^3$. Less formally said, we show how to reduce the powers of $u$ and $u^R$ in $x$ in such a way that the palindromic length remains bounded.

math.CO↗

arXiv2020

Palindromic Length of Words with Many Periodic Palindromes

The palindromic length $\text{PL}(v)$ of a finite word $v$ is the minimal number of palindromes whose concatenation is equal to $v$. In 2013, Frid, Puzynina, and Zamboni conjectured that: If $w$ is an infinite word and $k$ is an integer such that $\text{PL}(u)\leq k$ for every factor $u$ of $w$ then $w$ is ultimately periodic. Suppose that $w$ is an infinite word and $k$ is an integer such $\text{PL}(u)\leq k$ for every factor $u$ of $w$. Let $Ω(w,k)$ be the set of all factors $u$ of $w$ that have more than $\sqrt[k]{k^{-1}\vert u\vert}$ palindromic prefixes. We show that $Ω(w,k)$ is an infinite set and we show that for each positive integer $j$ there are palindromes $a,b$ and a word $u\in Ω(w,k)$ such that $(ab)^j$ is a factor of $u$ and $b$ is nonempty. Note that $(ab)^j$ is a periodic word and $(ab)^ia$ is a palindrome for each $i\leq j$. These results justify the following question: What is the palindromic length of a concatenation of a suffix of $b$ and a periodic word $(ab)^j$ with "many" periodic palindromes? It is known that $\lvert\text{PL}(uv)-\text{PL}(u)\rvert\leq \text{PL}(v)$, where $u$ and $v$ are nonempty words. The main result of our article shows that if $a,b$ are palindromes, $b$ is nonempty, $u$ is a nonempty suffix of $b$, $\vert ab\vert$ is the minimal period of $aba$, and $j$ is a positive integer with $j\geq3\text{PL}(u)$ then $\text{PL}(u(ab)^j)-\text{PL}(u)\geq 0$.

cs.FL↗

arXiv2020

Transition Property for $α$-Power Free Languages with $α\geq 2$ and $k\geq 3$ Letters

In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $α$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $α$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let $Σ_k$ denote an alphabet with $k$ letters. Let $L_{k,α}$ denote the $α$-power free language over the alphabet $Σ_k$, where $α$ is a rational number or a rational "number with $+$". If $α$ is a "number with $+$" then suppose $k\geq 3$ and $α\geq 2$. If $α$ is "only" a number then suppose $k=3$ and $α>2$ or $k>3$ and $α\geq 2$. We show that: If $u\in L_{k,α}$ is a right extendable word in $L_{k,α}$ and $v\in L_{k,α}$ is a left extendable word in $L_{k,α}$ then there is a (transition) word $w$ such that $uwv\in L_{k,α}$. We also show a construction of the word $w$.

cs.DM↗

arXiv2019

Upper bound for the number of closed and privileged words

A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if $\vert u\vert\leq 1$ or if $u$ contains a privileged border $w$ that appears exactly twice in $u$. Peltomäki (2016) presented the following open problem: "Give a nontrivial upper bound for $B(n)$", where $B(n)$ denotes the number of privileged words of length $n$. Let $D(n)$ denote the number of closed words of length $n$. Let $q>1$ be the size of the alphabet. We show that there is a positive real constant $c$ such that \[D(n)\leq c\ln{n}\frac{q^{n}}{\sqrt{n}}\mbox{, where }n>1\mbox{.}\] Privileged words are a subset of closed words, hence we show also an upper bound for the number of privileged words.

cs.DM↗

arXiv2019

A Unique Extension of Rich Words

A word $w$ is called rich if it contains $| w|+1$ palindromic factors, including the empty word. We say that a rich word $w$ can be extended in at least two ways if there are two distinct letters $x,y$ such that $wx,wy$ are rich. Let $R$ denote the set of all rich words. Given $w\in R$, let $K(w)$ denote the set of all words such that if $u\in K(w)$ then $wu\in R$ and $wu$ can be extended in at least two ways. Let $ω(w)=\min\{| u| \mid u\in K(w)\}$ and let $ϕ(n)=\max\{ω(w)\mid w\in R\mbox{ and }| w|=n\}$, where $n>0$. Vesti (2014) showed that $ϕ(n)\leq 2n$. In other words, it says that for each $w\in R$ there is a word $u$ with $| u|\leq 2| w|$ such that $wu\in R$ and $wu$ can be extended in at least two ways. We prove that $ϕ(n)\leq n$. In addition we prove that for each real constant $c>0$ and each integer $m>0$ there is $n>m$ such that $ϕ(n)\geq (\frac{2}{9}-c)n$. The results hold for each finite alphabet having at least two letters.

cs.DM↗

arXiv2019

Upper Bound for Palindromic and Factor Complexity of Rich Words

A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word (finite or infinite) over an alphabet with $q>1$ letters, let $F(w,n)$ be the set of factors of length $n$ of the word $w$, and let $F_p(w,n)\subseteq F(w,n)$ be the set of palindromic factors of length $n$ of the word $w$. We present several upper bounds for $| F(w,n)|$ and $| F_p(w,n)|$, where $w$ is a rich word. In particular we show that \[| F(w,n)| \leq (q+1)8n^2(8q^{10}n)^{\log_2{2n}}+q\mbox{.}\] In 2007, Bal{á}{\v z}i, Mas{á}kov{á}, and Pelantov{á} showed that \[| F_p(w,n)| +| F_p(w,n+1)| \leq | F(w,n+1)|-| F(w,n)|+2\mbox{,}\] where $w$ is an infinite word whose set of factors is closed under reversal. We generalize this inequality for finite words.

math.CO↗

arXiv2019

Construction Of A Rich Word Containing Given Two Factors

A finite word $w$ with $\vert w\vert=n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called \emph{rich}. Let $\Factor(w)$ be the set of factors of the word $w$. It is known that there are pairs of rich words that cannot be factors of a common rich word. However it is an open question how to decide for a given pair of rich words $u,v$ if there is a rich word $w$ such that $\{u,v\}\subseteq \Factor(w)$. We present a response to this open question:\\ If $w_1, w_2,w$ are rich words, $m=\max{\{\vert w_1\vert,\vert w_2\vert\}}$, and $\{w_1,w_2\}\subseteq \Factor(w)$ then there exists also a rich word $\bar w$ such that $\{w_1,w_2\}\subseteq \Factor(\bar w)$ and $\vert \bar w\vert\leq m2^{k(m)+2}$, where $k(m)=(q+1)m^2(4q^{10}m)^{\log_2{m}}$ and $q$ is the size of the alphabet. Hence it is enough to check all rich words of length equal or lower to $m2^{k(m)+2}$ in order to decide if there is a rich word containing factors $w_1,w_2$.

math.CO↗

arXiv2017

Bijections in de Bruijn Graphs

A T-net of order $m$ is a graph with $m$ nodes and $2m$ directed edges, where every node has indegree and outdegree equal to $2$. (A well known example of T-nets are de Bruijn graphs.) Given a T-net $N$ of order $m$, there is the so called "doubling" process that creates a T-net $N^*$ from $N$ with $2m$ nodes and $4m$ edges. Let $|X|$ denote the number of Eulerian cycles in a graph $X$. It is known that $| N^*|=2^{m-1}|N|$. In this paper we present a new proof of this identity. Moreover we prove that $|N|\leq 2^{m-1}$. Let $Θ(X)$ denote the set of all Eulerian cycles in a graph $X$ and $S(n)$ the set of all binary sequences of length $n$. Exploiting the new proof we construct a bijection $Θ(N)\times S(m-1)\rightarrow Θ(N^*)$, which allows us to solve one of Stanley's open questions: we find a bijection between de Bruijn sequences of order $n$ and $S(2^{n-1})$.

math.CO↗