Related papers: On Pansiot Words Avoiding 3-Repetitions
We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1) \leq \Delta C(n) + 2, for all n \in N. For a large class of words it is a better estimate…
A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this…
Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite…
In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate…
Form a random k-SAT formula on n variables by selecting uniformly and independently m=rn clauses out of all 2^k (n choose k) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r_k…
We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a…
We have primarily obtained three results on numbers of the form $p + 2^k$. Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form $p + 2^k$, disproving a conjecture by Erd\H{o}s as Chen…
A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…
In 2007, Grytczuk conjecture that for any sequence $(\ell_i)_{i\ge1}$ of alphabets of size $3$ there exists a square-free infinite word $w$ such that for all $i$, the $i$-th letter of $w$ belongs to $\ell_i$. The result of Thue of 1906…
If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at…
In this paper, we study the pattern occurrence in $k$-ary words. We prove an explicit upper bound on the number of $k$-ary words avoiding any given pattern using a random walk argument. Additionally, we reproduce several already known…
A word is called $\beta$-free if it has no factors of exponent greater than or equal to $\beta$. The repetition threshold $\mathrm{RT}(k)$ is the infimum of the set of all $\beta$ such that there are arbitrarily long $k$-ary $\beta$-free…
We consider the random $k$-SAT problem with $n$ variables, $m=m(n)$ clauses, and clause density $\alpha=\lim_{n\to\infty}m/n$ for $k=2,3$. It is known that if $\alpha$ is small enough, then the random $k$-SAT problem admits a solution with…
Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in…
In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxx^R was intermediate between polynomial and exponential. We now show that the same holds for the growth of the number of…
We characterize the clustering of a word under the Burrows-Wheeler transform in terms of the resolution of a bounded number of bispecial factors belonging to the language generated by all its powers. We use this criterion to compute, in…
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called…
The tight upper bound on the state complexity of the reverse of R-trivial and J-trivial regular languages of the state complexity n is 2^{n-1}. The witness is ternary for R-trivial regular languages and (n-1)-ary for J-trivial regular…
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares…
In combinatorics on words, a word w of length n over an alphabet of size q is said to be privileged if n <= 1 or if n >= 2 and w has a privileged border that occurs exactly twice in w. Forsyth, Jayakumar and Shallit proved that there exist…