Parity Index of Binary Words and Powers of Prime Words

Abstract

Let $f$ be a binary word and let ${\cal F}_d(f)$ 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 ${\cal F}_d(f)$. 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 $0^r1^s0^t$, $r,s,t \geq 1$. The conjecture has also been verified by computer for all words $f$ of length at most 10 and all $d\le 31$.