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Combinatorics on words studies finite and infinite sequences of symbols from an alphabet, especially repetition, ordering and occurrence of shorter words. A finite word is an element of the free monoid generated by its alphabet, with concatenation as its operation. For example, abab has two occurrences of ab and is the square of ab; aba is a palindrome. These questions connect algorithms on words with generating functions and symbolic descriptions of dynamical systems.

Jeffrey Shallit studies combinatorics on words, formal languages, automata theory and algorithmic number theory. His work connects finite descriptions of sequences with arithmetic structure and algorithms for deciding properties of those sequences.

His personal page gives a useful caution about experimentation: an apparently valid linear recurrence for one integer sequence fails only at index 11056. Long numerical agreement can motivate a conjecture, but a mathematical proof needs a structural reason that the identity continues indefinitely.^[ref]