On the Combinatorics of Palindromes and Antipalindromes

We prove a number of results on the structure and enumeration of palindromes and antipalindromes. In particular, we study conjugates of palindromes, palindromic pairs, rich words, and the counterparts of these notions for antipalindromes.Comment: 13 pages/ submitted to DLT 201

Constructions of words rich in palindromes and pseudopalindromes

A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.Comment: 26 page

On the Zero Defect Conjecture

peer reviewe

Palindromic richness for languages invariant under more symmetries

For a given finite group $G$ consisting of morphisms and antimorphisms of a free monoid $\mathcal{A}^*$, we study infinite words with language closed under the group $G$. We focus on the notion of $G$-richness which describes words rich in generalized palindromic factors, i.e., in factors $w$ satisfying $\Theta(w) = w$ for some antimorphism $\Theta \in G$. We give several equivalent descriptions which are generalizations of know characterizations of rich words (in the terms of classical palindromes) and show two examples of $G$-rich words