Total palindrome complexity of finite words

Abstract

a b s t r a c tThe palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).

Unravelling the biodiversity of nanoscale signatures of spider silk fibres

“…15). Palindromes are a ubiquitous feature of linguistics 43 , music 44 , mathematics 45 , computation 46 and biology 47,48 . The recurrence of palindromes in proteins represents a fundamental paradigm shift in structure-function studies because typical examples exhibit some kind of extraordinary feature 48 or undergo structural distortion under singular conditions 49 .…”

Generalized de Bruijn Words and the State Complexity of Conjugate Sets

“…However, Lemma 3 of that paper actually implies the existence of our generalized (circular) de Bruijn words of every length over a binary alphabet, although this was not stated explicitly. Anisiu, Blázsik, and Kása [2] discussed a related concept: namely, those length-N words w for which max 1≤i≤N ρ i (w) = max x∈Σ N k max 1≤i≤N ρ i (x) where ρ i (w) denotes the number of distinct length-i factors of w (here considered in the ordinary sense, not circularly). Also see [7].…”

Maximal State Complexity and Generalized de Bruijn Words

“…However, Lemma 3 of that paper actually implies the existence of our generalized (circular) de Bruijn words of every length over a binary alphabet, although this was not stated explicitly. Anisiu, Blázsik, and Kása [22] discussed a related concept: namely, those length-N words w for which max 1≤i≤N ρ i (w) = max x∈Σ N k max 1≤i≤N ρ i (x) where ρ i (w) denotes the number of distinct length-i factors of w (here considered in the ordinary sense, not circularly). Also see [23].…”