Abstract The factor complexity $$\mathcal {C}_{{\textbf {u}}}$$ C u of a sequence $$\textbf{u}= \varvec{u}_{0}\varvec{u}_{1}\varvec{u}_{2} \cdots $$ u = u 0 u 1 u 2 ⋯ over a finite alphabet counts the number of factors of length n occurring in $$\textbf{u}$$ u , i.e., $$\mathcal {C}_\textbf{u}{(n)} = \#{\mathcal {L}}_n(\textbf{u})$$ C u ( n ) = # L n ( u ) , where $${\mathcal {L}}_n\varvec{(}\textbf{u}\varvec{)}= {\{}\varvec{u}_{{i}}\varvec{u}_{{i+1}}\cdots \varvec{u}_{{i+n-1}}{: i} \in \mathbb {N}{\}}$$ L n ( u ) = { u i u i + 1 ⋯ u i + n - 1 : i ∈ N } . Two factors of $${\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}$$ L n ( u ) are said to be equivalent if they are equal or one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity $${r}_\textbf{u}$$ r u which counts the number of non-equivalent factors of $${\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}$$ L n ( u ) . They formulated the following conjecture: a sequence $$\textbf{u}$$ u <

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 $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $\varphi$ in the class there exists a palindrome $w$ such that $\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness.

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number β such that |β| > 1, and the digit set A is a finite set of real or complex digits (including 0). In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that β is any real or complex number, and digits are real or complex. We show that if (β, A) satisfies the so-called (OL) Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base β and alphabet A of contiguous integers, the system (β, A) has the (OL) Property if #A > |β| . Provided that addition and subtraction are realizable in parallel in the system (β, A), our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base β = 3+√5/2 with alphabet A = {-1, 0, 1}; base β = 2i with alphabet A = {-2, -1, 0, 1, 2} (redundant Knuth numeration system); and base β = -3/2 + z√3/2 = -1 + ω, where ω = exp 2iπ/3 , with alphabet A = {0, ±1, ±ω, ±ω2} (redundant Eisenstein numeration system).

The m-bonacci word is a generalization of the Fibonacci word to the m-letter alphabet 𝒜 = {0, ... ,m − 1}. It is the unique fixed point of the Pisot–type substitution �m : 0 → 01, 1 → 02, ... , (m − 2) → 0(m − 1), and (m − 1) → 0. A result of Adamcz