Related papers: Maximal right smooth extension chains
A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantov\'a and Starosta (Discrete Math. 313 (2013)) proved…
Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a…
Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p $\in$ (0, 1), and repeats some preceding word with complementary probability 1 -- p. More precisely, given a…
Let the root of the word $w$ be the smallest prefix $v$ of $w$ such that $w$ is a prefix of $vvv...$. $per(w)$ is the length of the root of $w$. For any $n\ge5$, an $n$-ary threshold word is a word $w$ such that for any factor (subword) $v$…
A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with…
Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric)…
A square is a word of the form $xx$ for a non-empty word $x$. Brlek and Li [Comb. Theory, 2025] proved that the number of distinct squares in a word $w$ of length $n$ is at most $n - \sigma$, where $\sigma$ is the number of letters used in…
A q-ary linear code of dimension k is called a maximum weight spectrum (MWS) code if it has the maximum possible number (viz. (q^k-1)/(q-1)) of different non-zero weights. We construct MWS codes from quasi-minimal codes, thus obtaining of…
A string $w$ is called a minimal absent word (MAW) for a string $S$ if $w$ does not occur as a substring in $S$ and all proper substrings of $w$ occur in $S$. MAWs are well-studied combinatorial string objects that have potential…
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 study the use of sampling for efficiently mining the top-K frequent itemsets of cardinality at most w. To this purpose, we define an approximation to the top-K frequent itemsets to be a family of itemsets which includes (resp., excludes)…
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…
A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper we introduce the definition of a \emph{maximal closed substring} (MCS), which is an occurrence of a closed substring that cannot be…
A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length $n$. The famous conjecture…
In this paper we investigate the problem of detecting, counting, and enumerating (generating) all maximum length plateau-$k$-rollercoasters appearing as a subsequence of some given word (sequence, string), while allowing for plateaus. We…
The piecewise complexity $h(u)$ of a word is the minimal length of subwords needed to exactly characterise $u$. Its piecewise minimality index $\rho(u)$ is the smallest length $k$ such that $u$ is minimal among its order-$k$ class $[u]_k$…
For $\alpha\geq 1$, an $\alpha$-gapped repeat in a word $w$ is a factor $uvu$ of $w$ such that $|uv|\leq \alpha |u|$; the two factors $u$ in such a repeat are called arms, while the factor $v$ is called gap. Such a repeat is called maximal…
For a stationary stochastic process $\{X_n\}$ with values in some set $A$, a finite word $w \in A^K$ is called a memory word if the conditional probability of $X_0$ given the past is constant on the cylinder set defined by $X_{-K}^{-1}=w$.…
This paper offers two elementary yet precise derivations of an exact formula \[ W(n) = \sum_{i=1} ^{n} \lceil \lg i \rceil = n \lceil \lg n \rceil - 2^{\lceil \lg n \rceil} + 1 \] for the maximum number $ W(n) $ of comparisons of keys…
Let $s_n$ be the number of words consisting of the ternary alphabet consisting of the digits 0, 1, and 2 such that no subword (or factor) is a square (a word concatenated with itself, e.g., $11$, $1212$, or $102102$). From computational…