Wild partitions and number theory

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,nx 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.