## 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}(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(Xn = r) = _{k}^{1}_{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} converges to a limit independent of r, and we 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 nonproduct 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.

Original language | English |
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Pages (from-to) | 1011-1038 |

Number of pages | 28 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved.

## Keywords

- Finate automata
- Limit theorems
- Pattern occurrences
- Random words
- Stanley-Wilf type limits
- Weak avoidance

## ASJC Scopus subject areas

- General Mathematics