On the Entropy and Letter Frequencies of Ternary Square-Free Words

We enumerate ternary length-$\ell$ square-free words, which are words avoiding squares of all words up to length $\ell$, for $\ell\le 24$. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary square-free words. We then consider ternary square-free words with fixed letter densities, thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary square-free words.