Abstract

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λ_{Q,n} counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function Λ_{Q}(x) = ∑ λ_{Q,n}x^{n} and use it to show that λ_{Q,n}^{1/n} tends to Q^{1/(p−1)}. We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture contrasts sharply with known results for function fields, and our arguments explain this contrast.