Abstract: We are all familiar with the representation of real numbers in an integer base b with digits 0, …, b-1. What happens if the base is not an integer? What happens if we change the set of permissible digits? What happens if that set differs for each digit? What if the digits need not be integers and are themselves represented in the non-integer base? And what have these questions to do with the growth rates of permutation classes?

Vatter proved that there are permutation classes of every growth rate above 2.481873. We will consider the connection with expansions in non-integer bases and explore how it was used to improve on this result by showing that the set of permutation class growth rates contains every value at least 2.356984 and also includes a sequence of intervals whose infimum is approximately 2.355257.