Finite automata and pattern avoidance in words

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 \cite{Reg1998} for exact asymptotics for the number of words on $k$ letters of length $n$ that avoids the pattern $\mathrm{12...}(\mathrm{\ell }+1)$. Moreover, we give the first combinatorial proof of the exact formula \cite{Burstein} for the number of words on $k$ letters of length $n$ avoiding a three letter permutation pattern.