Related papers: Improved Upper Bounds on all Maximal $\alpha$-gapp…

For odd prime numbers $p < q$, let $\Phi_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\Phi_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder…

It is shown that the maximum number of patterns that can occur in a permutation of length $n$ is asymptotically $2^n$. This significantly improves a previous result of Coleman.

The aim of the present paper is to prove that the maximum number of edges in a 3-uniform hypergraph on n vertices and matching number s is max{\binom(3s+2,3), \binom(n,3) - \binom(n-s,3)} for all n,s, n >= 3s+2.

We show in this note that the average number of terms in the optimal double-base number system is in Omega(n / log n). The lower bound matches the upper bound shown earlier by Dimitrov, Imbert, and Mishra (Math. of Comp. 2008).

We show that for any two distinct words $ s_1, s_2 $ over an arbitrary alphabets, there exists a deterministic finite automaton with $ O(\log^2 n) $ states that accepts $ s_1 $ and rejects $ s_2 $. This improves the previous upper bound of…

A run is a maximal occurrence of a repetition $v$ with a period $p$ such that $2p \le |v|$. The maximal number of runs in a string of length $n$ was studied by several authors and it is known to be between $0.944 n$ and $1.029 n$. We…

We prove that every concatenation of $10$ or more binary squares contains an overlap. The bound $10$ is best possible. In contrast, over a ternary alphabet, there are infinitely long overlap-free words that consist of a concatenation of…

Let $\Gamma$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(\Gamma,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$…

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The…

We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations…

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally…

We characterize the formulas that are avoided by every $\alpha$-free word for some $\alpha>1$. We study the avoidability index of formulas whose fragments are of the form $XYX$. The largest avoidability index of an avoidable palindrome…

For a word $\pi$ and integer $i$, we define $L^i(\pi)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(\pi):=\max_i L^i(\pi)$. In this paper we estimate the expected values of $L^1(\pi)$ and $L(\pi)$…

We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with $n$ faces admits, up to constants, an…

A word $w$ is called synchronizing (recurrent, reset, magic, directable) word of deterministic finite automaton (DFA) if $w$ sends all states of the automaton to a unique state. In 1964 Jan \v{C}erny found a sequence of n-state complete DFA…

Mignosi, Restivo, and Salemi (1998) proved that for all $\epsilon > 0$ there exists an integer $N$ such that all prefixes of the Fibonacci word of length $\geq N$ contain a suffix of exponent $\alpha^2-\epsilon$, where $\alpha =…

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound log_{2}(n)} + 1 for the number of distinct Lyndon factors that a Lyndon…

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the…

Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap $g$ (maximum of the…