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Suppose $p_k(n)$ is the number of partitions of the integer $n$ into $k$ parts. For example, the partition of 5 into 2 parts is; $p_2(5) = 2$, since the partitions are (4,1) and (3,2).

What is a good combinatorial proof that $p_k(n) \leq (n-k+1)^{k-1}$ for $k \leq n$? Or must this be done algebraically?

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  • $\begingroup$ what's $p_k(n)$? $\endgroup$ – Phicar Sep 21 '17 at 4:16
  • $\begingroup$ Updated the question to include the definition. $\endgroup$ – bashmike Sep 21 '17 at 4:20
  • $\begingroup$ $p_k(n)$ is the Stirling number of second kind. $\endgroup$ – GAVD Sep 21 '17 at 4:35
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    $\begingroup$ @GAVD No, it isn't. The Stirling number of the second kind counts partitions of sets, $p_k(n)$ counts partitions of numbers. $\endgroup$ – bof Sep 21 '17 at 5:43
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Here is an extremely straightforward way to see this:

A $k$-partition of $n$ is uniquely determined by the first $k-1$ values. Each element of a $k$-partition must be at most $n-k+1$, since the other elements are positive integers and sum to $\ge k-1$. Therefore the number of partitions of $n$ into $k$ parts is no larger than the number of $(k-1)$-tuples of integers between $1$ and $n-k+1$.

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