Uncountably many enumerations of well-quasi-ordered permutation classes

TL;DR

This paper constructs an uncountable family of well-quasi-ordered permutation classes with distinct enumeration sequences, challenging previous conjectures about their generating functions and showing many lack D-finite or D-algebraic properties.

Contribution

It introduces a novel uncountable collection of permutation classes with unique enumeration sequences, disproving the conjecture that all such classes have algebraic generating functions.

Findings

Constructed uncountably many well-quasi-ordered permutation classes.

Showed these classes have diverse enumeration sequences.

Proved many classes lack D-finite or D-algebraic generating functions.

Abstract

We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and in fact shows that many such classes lack D-finite or D-algebraic generating functions. Our construction is based on an uncountably large collection of factor-closed, well-quasi-ordered binary languages due to Pouzet.