Combinatorics on Words

Words (strings of symbols) are fundamental in computer processing. Indeed, each bit of data processed by a computer is a string, and nearly all computer software use algorithms on strings. There are also abundant supply of applications of these algorithms in other areas such as data compression, DNA sequence analysis, computer graphics, cryptography, and so on. Combinatorics on words belongs to discrete mathematics and theoretical computer science, and it is, also historically, close to many branches of mathematics and computer science. Related areas in discrete mathematics include automata, formal languages, probability, semigroups, groups, dynamical systems, combinatorial topology and number theory. The history of combinatorics on words goes back almost 100 years to Axel Thue and his work on repetitions in words. Systematic study of combinatorial properties on words was initiated in late 1950s by Marcel Schützenberger. Good overviews of the present state of art can be found in the books of Lothaire [11, 12] and in the survey chapter [5] by Choffrut and Karhumäki in the Handbook of Formal Languages. The generic topic of the combinatorics on words is to study general properties of words (strings or sequences of discrete events), as well as sets, functions and sequences of words. The theory covers both finite and infinite words. In these lectures we concentrate mostly on (special problems of) finite words. Combinatorics on words has the Mathematical Reviews classification 68R15.