Young classes of permutations

TL;DR

This paper characterizes permutation classes based on tableau shape properties, linking them to dominance order and establishing a bound on containing both increasing and decreasing sequences.

Contribution

It provides a novel characterization of permutation classes with shape-based properties using dominance order and sequence length bounds.

Findings

Characterization of classes based on tableau shape properties

Existence of a constant k bounding sequence lengths in the class

Connection between shape properties and permutation containment

Abstract

We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance order for partitions (and their conjugates) and shows that for any such class there is a constant k such that no permutation in the class can contain both an increasing and a decreasing sequence of length k.