Parity Index of Binary Words and Powers of Prime Words

Let f be a binary word and let Fd(f) be the set of words of length d which do not contain f as a factor (alias words that avoid the pattern f). A word is called even/odd if it contains an even/odd number of 1s. The parity index of f (of dimension d) is introduced as the difference between the number of even words and the number of odd words in Fd(f). A word f is called prime if every nontrivial suffix of f is different from the prefix of f of the same length. It is proved that if f is a power of a prime word, then the absolute value of the parity index of f is at most 1. We conjecture that no other word has this property and prove the conjecture for words 010, r, s, t C 1. The conjecture has also been verified by computer for all words f of length at most 10 and all d B 31.