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Let's $k_1+\ldots +k_p=1$. What functions of class $c^k$ are upper bounds for multinomial coeficientes $$ \begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix}=\frac{n}{k_1!\cdot k_2!\cdot\ldots\cdot k_p!} \quad? $$ More precisely:

Question: Fix $k\in\{0,1,\ldots,\infty\}$. There is a functions $f:\mathbb{R}\to\mathbb{R}$, $g:\mathbb{R}\to\mathbb{R}$ and $F:\mathbb{R}^p\to\mathbb{R}$ of class $C^k$ such that $$ \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} =f(n)> \begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} \neq \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} $$ $$ \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} =g(\max\{k_1,\ldots,k_p\})\geq \begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix}\neq \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} $$ $$ \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} =F(k_1,\ldots,k_p)\geq \begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix}\neq \max_{k_1,\ldots,k_p}\begin{pmatrix} n\\k_1,\ldots,k_p\end{pmatrix} $$ and $$\lim_{t\to\infty}f(t)=0,$$ $$\lim_{t\to\infty} g(t)=0$$ $$\lim_{\min\{k_1,\cdots,k_p\}\to\infty}F(k_1,\cdots,k_p)=0 \quad ?$$

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