Permutations with exactly one copy of a decreasing pattern of length k

TL;DR

This paper constructs a specific injection between permutations with exactly one decreasing pattern of length k and pattern-avoiding permutations, proving the generating function's nonrationality and nonalgebraicity for certain k.

Contribution

It introduces a novel injection linking permutations with a single decreasing pattern to pattern-avoiding permutations and establishes the nonrationality and nonalgebraicity of their generating functions.

Findings

The generating function is not rational.

For even k ≥ 4, the generating function is not algebraic.

The injection extends to a broader class of patterns.

Abstract

We construct an injection from the set of permutations of length $n$ that contain exactly one copy of the decreasing pattern of length $k$ to the set of permutations of length $n+2$ that avoid that pattern. We then prove that the generating function counting the former is not rational, and in the case when $k$ is even and $k\geq 4$, it is not even algebraic. We extend our injection and our nonrationality result to a larger class of patterns.