Is there any nontrivial upper bound for the size of $\{(x_1,\dots,x_k)\in\mathbb{N}^k:\sum_{i=1}^kx_i=n\}?$ We may assume $k,n$ are large enough but $k\ll n$, so an asymptotic bound is also helpful.
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$\begingroup$ This can be computed exactly as ${{n+k-1}\choose {k-1}}$ or ${{n+k-1}\choose n}$, if that helps at all. $\endgroup$– kccuCommented May 11, 2017 at 17:19
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$\begingroup$ Oh, I see. By adding $k,$ it converges to the problem that $k$ positive integers summing up to $n.$ $\endgroup$– ConnorCommented May 11, 2017 at 17:20
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$\begingroup$ Exactly. ${}{}{}$ $\endgroup$– kccuCommented May 11, 2017 at 17:22
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