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\begin{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 $\lambda_{Q,n}$ counting the number of certain  
wild
partitions of $n$.  We give an explicit formula for the 
  corresponding generating function $\Lambda_Q(x) = \sum \lambda_{Q,n} x^n$ and use
  it to show that  $\lambda^{1/n}_{Q,n}$ 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.  
\end{abstract}

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