Counting Words by Number of Occurrences of Some Patterns

Abstract

We give asymptotic expressions for the number of words containing a given number of occurrences of a pattern for two families of patterns with two parameters each. One is the family of classical patterns in the form $22\cdots 212 \cdots 22$ and the other is a family of partially ordered patterns. The asymptotic expressions are in terms of the number of solutions to an equation, and for one subfamily this quantity is the number of integer partitions into $q$th order binomial coefficients.