We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of -adic fields. For a power of , we get a sequence of numbers counting the number of certain wild partitions of . We give an explicit formula for the corresponding generating function and use it to show that tends to . 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.