Distinct Palindromes Research Articles

Critical Exponent of Binary Words with Few Distinct Palindromes

We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with 18 and 20 palindromes happen to be morphic images of the fixed point of the morphism $\texttt{0}\mapsto\texttt{01}$, $\texttt{1}\mapsto\texttt{21}$, $\texttt{2}\mapsto\texttt{0}$.

Tight Bound for the Number of Distinct Palindromes in a Tree

For an undirected tree with edges labeled by single letters, we consider its substrings, which are labels of the simple paths between two nodes. A palindrome is a word $w$ equal to its reverse $w^R$. We prove that the maximum number of distinct palindromic substrings in a tree of $n$ edges satisfies $\text{pal}(n)=O(n^{1.5})$. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who showed that $\text{pal}(n)=\Omega(n^{1.5})$. Hence, we settle the tight bound of $\Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for trees that are simple paths, the maximum palindromic complexity is exactly $n+1$.
 We also propose an $O(n^{1.5} \log^{0.5}{n})$-time algorithm reporting all distinct palindromes and an $O(n \log^2 n)$-time algorithm finding the longest palindrome in a tree.

On Morphisms Preserving Palindromic Richness

It is known that each word of length n contains at most n + 1 distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric d-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class Pret. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism ϕ in the class there exists a palindrome w such that ϕ(a)w is a first complete return word to w for each letter a. We characterize Pret morphisms which preserve richness over a binary alphabet. We also study marked Pret morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class Pret and that it preserves richness.

On k-abelian palindromes

A word is called a palindrome if it is equal to its reversal. In the paper we consider a k-abelian modification of this notion. Two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. We say that a word is a k-abelian palindrome if it is k-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length n can contain at most n+1 distinct palindromes as its factors; such words are called rich. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called poor. We show that in the k-abelian case there exist infinite words containing finitely many distinct k-abelian palindromic factors. For rich words we show that there exist finite words of length n containing Θ(n2) distinct k-abelian palindromes as their factors.

Palindromic language of thin discrete planes

We work on the Réveillès hyperplane P(v,0,ω) with normal vectorv∈Rd, shiftμ=0 and thicknessω∈R. Such a hyperplane is connected as soon as ω is greater than some value Ω(v,0), called the connecting thickness of v with null shift. In the case where v satisfies the so called Kraaikamp and Meester criterion, at the connecting thickness the hyperplane has very specific properties. First of all the adjacency graph of the voxels forms a tree. This tree appeared in many works both in discrete geometry and in discrete dynamical systems. In addition, it is well known that for a finite coding of length n of discrete lines, the number of palindromes in the language is exactly n+1. We extend this notion of language to labeled trees and we compute the number of distinct palindromes. In fact for our voxel adjacency trees with n letters we show that the number of palindromes in the language is also n+1. This result establishes a first link between combinatorics on words, palindromic languages, voxel adjacency trees and connecting thickness of Réveillès hyperplanes. It also provides a better understanding of the combinatorial structure of discrete planes.

Palindromes in circular words

There is a very short and beautiful proof that the number of distinct non-empty palindromes in a word of length n is at most n. In this paper we show, with a very complicated proof, that the number of distinct non-empty palindromes with length at most n in a circular word of length n is less than 5n/3. For n divisible by 3 we present circular words of length n containing 5n/3−2 distinct palindromes, so the bound is almost sharp. The paper finishes with some open problems.

Counting distinct palindromes in a word in linear time

We design an algorithm to count the number of distinct palindromes in a word w in time O ( | w | ) , by adapting an algorithm to detect all occurrences of maximal palindromes in a given word and using the longest previous factor array. As a direct consequence, this shows that the palindromic richness (or fullness) of a word can be checked in linear time.

A new characteristic property of rich words

Originally introduced and studied by the third and fourth authors together with J. Justin and S. Widmer (2008), rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes. Several characterizations of rich words have already been established. A particularly nice characteristic property is that all ‘complete returns’ to palindromes are palindromes. In this note, we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix.