Order of weak M-relation and Parikh matrices

The classical inference problem on words studies whether a word can be uniquely inferred from some given binomial coefficients. The entries of the Parikh matrix are certain canonical binomial coefficients depending on the ordered alphabet. The wide open injectivity problem asks when two words are M-equivalent, that is, sharing the same Parikh matrix. Two words are weakly M-related if and only if they are M-equivalent with respect to some ordering on the alphabet. In this work, we introduce a function that measures how hard it is for two words to become weakly M-related. We characterize this function and solve various connected combinatorial questions.