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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 ( X n = r ) = 1 k n f r v ( k , n ) . In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence ( f r v ( k , n ) ) 1 ∕ n converges to a limit independent of r, and 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 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.

Finite Automata, Probabilistic Method, and Occurrence Enumeration of a Pattern in Words and Permutations