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In this note we provide a (decidable) graph-structural characterisation ofthe infiniteness of $L(w_1, ..., w_k)$, where $L(w_1, ..., w_k) = \{w \in A^* ||w|_{w_1} = \cdots = |w|_{w_k}\}$ is the set of all words that contain the samenumber of subword occurrences of parameter words $w_1, ..., w_k$. We alsoprovide the decidable characterisation of the equivalence for those languages.Although those two decidability results are also obtained from more generalknown decidability results on unambiguous constrained automata, this note triesto give a self-contained (without the knowledge about constrained automata)proof of the decidability.

Note on the Infiniteness and Equivalence Problems for Word-MIX Languages