2 Finite Automata and Pattern Avoidance

We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula [20] for exact asymptotics for the number of words on k letters of length n that avoids the pattern 12 · · · (ℓ + 1). Moreover, we give the first combinatorial proof of the exact formula [9] for the number of words on k letters of length n avoiding a three letter permutation pattern.