Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutations

The main theme of this paper is the enumeration of the order-isomorphic occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence f_r^v(k, n), the number of n-array k-ary words that contain a given pattern v exactly r times. In addition, we study the asymptotic behavior of the random variable Xn, the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity P(Xn = r) = _k¹_n f_r^v(k, n). In particular, we show that for any r \geq 0, the Stanley-Wilf sequence \bigl(f_r^v(k, n)^{\bigr) 1/n} converges to a limit independent of r, and we determine the value of the limit. We then obtain several limit theorems for the distribution of Xn, including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of Xn. Furthermore, we introduce a concept of weak avoidance and link it to a certain family of nonproduct measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.