---
abstract: 'We enumerate three specific permutation classes defined by two forbidden patterns of length four. The techniques involve inflations of geometric grid classes.'
author:
- |
    Michael H. Albert\
    Department of Computer Science\
    University of Otago\
    Dunedin, New Zealand\
    M. D. Atkinson\
    Department of Computer Science\
    University of Otago\
    Dunedin, New Zealand\
    Vincent Vatter[^1]\
    Department of Mathematics\
    University of Florida\
    Gainesville, Florida USA\
title: 'Inflations of Geometric Grid Classes: Three Case Studies'
---

Introduction {#infinite-simples-intro}
============

*Classes* of permutations are sets that are closed downwards under taking subpermutations. They are often presented as sets ${\mathcal{C}}$ that avoid a given set $B$ of permutations (i.e. the members of ${\mathcal{C}}$ have no subpermutation in the set $B$). We express this by the notation ${\mathcal{C}}={\operatorname{Av}}(B)$. We may take $B$ to be an antichain (a set of pairwise incomparable permutations), in which case we say that $B$ is the *basis* of ${\mathcal{C}}$.

Much of the inspiration for the early work on permutation classes was driven by the enumeration problem: given ${\mathcal{C}}={\operatorname{Av}}(B)$, how many permutations of each length does ${\mathcal{C}}$ contain? The answer to such a question could be a formula giving this number $|{\mathcal{C}}_n|$ in terms of the length, $n$, a generating function 