Permutation Patterns and Continued Fractions

Abstract

We find, in the form of a continued fraction, the generating function for the number of $(132)$-avoiding permutations that have a given number of $(123)$ patterns, and show how to extend this to permutations that have exactly one $(132)$ pattern. We also find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.