Let $p_k(n)$ the number of partitions of $n$ into exactly $k$ parts. Do we have $p_k(n) \le 2^{n-k}$ ?
2026-03-26 01:23:26.1774488206
Upper bound for the number of partitions of $n$ into exactly $k$ parts
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The number of partitions of a set with $n$ elements into $k$ blocks is given by the Stirling number of the second kind $S(n,k)$. There is an explicit formula and a recursion formula for it.