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 , the number of -array -ary words that contain a given pattern exactly times. In addition, we study the asymptotic behavior of the random variable , the number of pattern occurrences in a random -array word. The two topics are closely related through the identity . In particular, we show that for any ≥ 0, the Stanley-Wilf sequence converges to a limit independent of , and determine the value of the limit. We then obtain several limit theorems for the distribution of , including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of . Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product 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.