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The main theme of this paper is the enumeration of the occurrence of apattern in words and permutations. We mainly focus on asymptotic properties ofthe sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain agiven pattern $v$ exactly $r$ times. In addition, we study the asymptoticbehavior of the random variable $X_n,$ the number of pattern occurrences in arandom $n$-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$r\geq 0,$ the Stanley-Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{1/n}$ convergesto a limit independent of $r,$ and determine the value of the limit. We thenobtain several limit theorems for the distribution of $X_n,$ including a CLT,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 certainfamily of non-product measures on words that penalize pattern occurrences butdo not forbid them entirely. We analyze this family of probability measures ina small parameter regime, where the distributions can be understood as aperturbation of a uniform measure. Finally, we extend some of our results forwords, including the one regarding the equivalence of the limits of theStanley-Wilf sequences, to pattern occurrences in permutations.

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