The number of inversions of a word: pairs of positions with . Unlike an inversion of a permutation, this definition permits repeated letters in the word.
For slope and intercept , the lower mechanical word has letters . These letters are zero or one. Replacing floors by ceilings defines the upper mechanical word. An irrational slope gives a Sturmian word.
The floor-based mechanical word .
A nonempty finite word that is strictly smaller in lexicographic order than each of its nontrivial cyclic rotations. For example, is a Lyndon word, whereas is not because its rotation is smaller.
A prefix of a word that is itself a Lyndon word. The longest Lyndon prefix is useful when ranking necklaces and substrings of a De Bruijn sequence.
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Combinatorics on words is a branch of combinatorial mathematics that deals with the study of words and sequences formed from a finite alphabet. It involves analyzing the properties, structures, and patterns of these sequences, exploring various aspects such as counting, arrangements, and combinatorial structures associated with words. This field intersects with other areas such as formal languages, automata theory, computer science, linguistics, and information theory.