On the growth of merges and staircases of permutation classes

TL;DR

This paper investigates the growth rates of permutation class merges, providing conditions under which the known upper bound is achieved, especially for principal classes, and offers new proofs for existing results.

Contribution

It introduces staircase techniques to determine when the maximum growth rate bound for merges of permutation classes is attained, extending to principal classes and rederiving known results.

Findings

Upper bound on growth rate often achieved in merges

Staircase methods provide sufficient conditions for maximum growth

Reproves a known result of Bóna using new techniques

Abstract

There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient conditions for this upper bound to be achieved. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques can be used to reprove a result of B\'ona.