On the Number of Closed Factors in a Word

Abstract

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special.We investigate the structure of closed factors of words. We show that a word of length n contains at least n + 1 distinct closed factors, and characterize those words having exactly n + 1 closed factors. Furthermore, we show that a word of length n can contain Θ(n²) many distinct closed factors

The sequence of open and closed prefixes of a Sturmian word

“…We now define the oc-sequence of a word. [2] · · · w[n] · · · be a finite or infinite word over Σ. We define oc(w) = c(1)c(2) · · · c(n) · · · , called the oc-sequence of w, as the binary sequence whose n-th element is 0 if the prefix of length n of w is open, or 1 if it is closed.…”

“…Let Σ be a finite nonempty set (the alphabet). A (finite) word w = w [1]w [2] · · · w[n] with w[i] ∈ Σ is closed (also known as periodic-like [6]) if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. For example, the words abca, ababa and aabaab are closed -any word of length 1 is closed, the empty word being a factor that occurs both as a prefix and as a suffix but does not have internal occurrences; the words ab, aab and aaba, instead, are open.…”

“…Given a finite or infinite word w = w [1]w [2] · · · , the sequence oc(w) of open/closed prefixes of w, that we refer to as the oc-sequence of w, is the binary sequence c(1)c(2) · · · whose n-th element is 1 if the prefix of w of length n is closed, 0 if it is open. For example, if w = abcab, then oc(w) = 10011.…”

“…An infinite word w over Σ is a sequence w : N + → Σ, written as w = w[1]w [2] · · · w[n] · · · . Prefixes and factors of infinite words are naturally defined, as is the product uw of a finite word u and an infinite word w. Let Σ ω denote the set of infinite words over Σ.…”

Closed factorization

“…Closed strings were introduced by Fici [1] as objects of combinatorial interest in the study of Trapezoidal and Sturmian words. Since then, Badkobeh, Fici, and Liptak [2] have proved a tight lowerbound for the number of closed factors (substrings) in strings of given length and alphabet size.…”