A permutation class is said to be deflatable if its simple permutations are contained within a proper subclass. Deflatable classes are often easier to describe, analyze, and enumerate than their non-deflatable counterparts. This paper presents theorems guaranteeing the non-deflatability of principal classes, constructs an infinite family of deflatable principal classes, and provides examples of each.
The authors are grateful to Vince Vatter for participating in discussions which furthered this research. In particular he was in part responsible for the original proof of Proposition 3.10 which convinced us that "except in trivial cases principal classes aren't deflatable" was perhaps not as obvious or as easy as one might initially think -- and indeed of course we now know it to be false. Additionally, Cheyne Homberger and Jay Pantone wish to thank Michael Albert and Mike Atkinson for their hospitality at the University of Otago in March and April of 2014.