Further Applications of a Power Series Method for Pattern Avoidance

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $x$ of $w$ and no non-erasing morphism $h$ from $\Delta^*$ to $\Sigma^*$ such that $h(p) = x$. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns $p$ there are exponentially many words of length $n$ over a $4$-letter alphabet that avoid $p$. We consider some further consequences of their work. In particular, we show that any pattern with $k$ variables of length at least $4^k$ is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth.