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

SIAM J Discret Math. 2020;34(2):1011-1038. doi: 10.1137/19m1262206. Epub 2020 Apr 8.

## Abstract

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 ${f}_{r}^{v}\left(k,n\right)$ , 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 $P\left({X}_{n}=r\right)=\frac{1}{{k}^{n}}{f}_{r}^{v}\left(k,n\right)$ . In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence ${\left({f}_{r}^{v}\left(k,n\right)\right)}^{1∕n}$ converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of 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.

Keywords: Stanley-Wilf type limits; finite automata; limit theorems; pattern occurrences; random words; weak avoidance.