Most principal permutation classes have nonrational generating functions

TL;DR

This paper demonstrates that for most permutation patterns, the generating functions counting pattern-avoiding permutations are nonrational, revealing complex combinatorial structures and growth behaviors.

Contribution

It proves that for most patterns, the generating functions are nonrational and establishes the supercritical nature of related functional relations.

Findings

Most permutation patterns have nonrational generating functions.

The relation between generating functions is typically not supercritical.

Pattern avoidance sequences exhibit complex, nonrational growth patterns.

Abstract

We prove that for any fixed $n$, and for most permutation patterns $q$, the number $\textup{Av}_{n,\ell}(q)$ of $q$-avoiding permutations of length $n$ that consist of $\ell$ skew blocks is a monotone decreasing function of $\ell$. We then show that this implies that for most patterns $q$, the generating function $\sum_{n\geq 0} \textup{Av}_n(q)z^n$ of the sequence $\textup{Av}_n(q)$ of the numbers of $q$-avoiding permutations is not rational. Placing our results in a broader context, we show that for rational power series $F(z)$ and $G(z)$ with nonnegative real coefficients, the relation $F(z)=1/(1-G(z))$ is supercritical, while for most permutation patterns $q$, the corresponding relation is not supercritical.