On the entropy and letter frequencies of ternary square-free words

Abstract

We enumerate ternary length-ℓ square-free words, which are words avoiding squares of all words up to length ℓ, for ℓ ≤ 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.