Palindromes in circular words (Q401312)

A palindrome is a word equal to its mirror image. It is well known that a word of length \(n\) contains at most \(n\) distinct non-empty palindromic factors. An elegant short proof can be found in the paper. A circular word is obtained from a usual word by ``glueing'' its ends together, or, equivalently, by writing the word in a counter-clockwise way around the circumference of a circle. Thus, circular words are not words in the usual sense. The number of palindromic factors in a circular word of length \(n\) can therefore be greater than \(n\). The current paper presents a lengthy and complex proof of the fact that this number is sharply smaller than \(5n/3\). Moreover, for each \(n\) there is a circular word of length \(n\) containing \(5n/3-2\) palindromic factors. There remains the open problem, in which direction the tiny gap can be closed.