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 X_n , the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity $P(X_n=r)=\frac{1}{k^n}{f}_r^v(k,n)$ . In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence ${\left(f_r^v\right(k,n\left)\right)}^{1∕n}$ converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of X_n , including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of X_n . 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.