Journal of Integer Sequences, Vol. 10 (2007), Article 07.6.6

Wild Partitions and Number Theory

David P. Roberts
Division of Science and Mathematics
University of Minnesota, Morris
Morris, MN, 56267


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.

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(Concerned with sequences A000041 A000085 A010054 A033687 A131139 and A131140 .)

Received March 28 2007; revised version received June 18 2007. Published in Journal of Integer Sequences June 18 2007.

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