Patterns and Fractions

TL;DR

This paper derives a continued fraction generating function for counting permutations avoiding certain patterns, extending to permutations with exactly one pattern, and explores properties similar to Ramanujan's continued fractions.

Contribution

It introduces a continued fraction representation for the generating function of (132)-avoiding permutations with a specified number of (123) patterns, including the case of exactly one (132) pattern.

Findings

Derived continued fraction for (132)-avoiding permutations with (123) patterns

Extended analysis to permutations with exactly one (132) pattern

Identified properties of the continued fraction related to Ramanujan's work

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 find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.