Related papers: Counting Lyndon factors
The mapping of lexical meanings to wordforms is a major feature of natural languages. While usage pressures might assign short words to frequent meanings (Zipf's law of abbreviation), the need for a productive and open-ended vocabulary,…
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…
Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property,…
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…
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A…
Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of…
The number of frequencies of factors of length $n+1$ in a recurrent aperiodic infinite word does not exceed $3\Delta \C(n)$, where $\Delta \C (n)$ is the first difference of factor complexity, as shown by Boshernitzan. Pelantov\'a together…
A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely…
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…
Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a $k$-letter alphabet. The obtained lower bounds for small values of…
A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal…
We prove that a random word of length $n$ over a $k$-ary fixed alphabet contains, on expectation, $\Theta(\sqrt{n})$ distinct palindromic factors. We study this number of factors, $E(n,k)$, in detail, showing that the limit…
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 generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as…
A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of…
Superpermutations are words over a finite alphabet containing every permutation as a factor. Finding the minimal length of a superpermutation is still an open problem. In this article, we introduce superpermutations matrices. We establish a…
To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same…
In the Shortest-Superstring problem, we are given a set of strings S and want to find a string that contains all strings in S as substrings and has minimum length. This is a classical problem in approximation and the best known…
If $u$ and $v$ are two words, the correlation of $u$ over $v$ is a binary word that encodes all possible overlaps between $u$ and $v$. This concept was introduced by Guibas and Odlyzko as a key element of their method for enumerating the…
In this note, we establish the convexity and monotonicity for affine standard Lyndon words in all types, generalizing the $A$-type results of arXiv:2305.16299. We also derive partial results on the structure of imaginary standard Lyndon…