Search results for "68R15"
showing 9 items of 19 documents
Abelian-Square-Rich Words
2017
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-sq…
On the least number of palindromes contained in an infinite word
2013
We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.
On the Lie complexity of Sturmian words
2022
Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic techniques, that the Lie complexity is bounded above by the first difference of the factor complexity plus one; hence, it is uniformly bounded for words with linear factor complexity, and, in particular, it is at most 2 for Sturmian words, which are precisely the words with factor complexity $n+1$ for every $n$. In this note, we provide an elementary combinatorial proof of the result of Bell and Shallit and give an exact formula for the Lie complexity of any Sturmi…
On the Structure of Bispecial Sturmian Words
2013
A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian wo…
Pattern statistics in faro words and permutations
2021
We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Fina…
Open and Closed Prefixes of Sturmian Words
2013
A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.
The Intersection of $3$-Maximal Submonids
2020
Very little is known about the structure of the intersection of two $k$-generated monoids of words, even for $k=3$. Here we investigate the case of $k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be non-trivially decomposed into at most $k$ words. We characterize the intersection in the case of two $3$-maximal monoids.
Some Remarks on Differentiable Sequences and Recursivity
2010
International audience; We investigate the recursive structure of differentiable sequences over the alphabet {1, 2}. We derive a recursive formula for the (n + 1)-th symbol of a differentiable sequence, which yields to a new recursive formula for the Kolakoski sequence. Finally, we show that the sequence of absolute differences of consecutive symbols of a differentiable sequence u is a morphic image of the run-length encoding of u.
Aperiodicity in finitely generated bi-ideals and bounded bi-ideals
2014
Disertācijā pētīts aperiodiskums divu vienpusēji bezgalīgu vārdu klašu – galīgi ģenerētu biideālu un ierobežotu bi-ideālu – kontekstā. Gan galīgi ģenerētus bi-ideālus, gan ierobežotus biideālus ir ērti uzdot, izmantojot bāzes virkni. Ierobežotu bi-ideālu gadījumā tiek nofiksēta galīga vārdu kopa, no kuras tiek uzģenerēta bāzes virkne. Savukārt galīgi ģenerētu bi-ideālu gadījumā bāzes virkne ir periodiska, līdz ar to var uzskatīt, ka ir dota galīga bāze. Efektīvā ģenerēšana un zināmie aperiodiskuma nosacījumi kalpo par motivāciju apskatīt galīgi ģenerētu bi-ideālu potenciālo pielietojumu kriptogrāfijā – aperiodisku gadījuma skaitļu ģenerēšanā. Savukārt, ierobežoti bi-ideāli ir galīgi ģenerēt…