In this paper we prove that for any infinite word w whose set of factors is closed under reversal. the following conditions are equivalent: (I) all complete returns to palindromes are palindromes: (II)P(n) + P(n + 1) - C(n) + 2 for all n. where P (resp. C) denotes the palindromic complexity (resp. factor complexity) function of w, which counts the number of distinct palindromic factors (resp. factors) of each length in w

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Abstract. In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the well distributed occurrences (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WELLDOC property if, for each factor w of u, positive integer m, and vector v ∈ Z d m , there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. (The Parikh vector of a finite word v over an alphabet A has its i-th component equal to the number of occurrences of the i-th letter of A in v.) We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 [12] and PractRand [6] statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property.

Originally introduced and studied by the third and fourth authors together with J. Justin and S. Widmer in arXiv:0801.1656, 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.Comment: 6 page

Trapezoidal words are words having at most n+1 distinct factors of length n for every n≥0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, semicentral words, and show that they are characterized by the property that they can be written as uxyu, for a central word u and two different letters x,y. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence