Consider the following sets in $\mathbb{N}^{n}$ $$ F_{n} := \{ (x_1,..,x_n) \in \mathbb{N}^{n}: \sum_{i=1}^{i=n}x_i \ge A_n, \quad \max_{1\le i \le n}x_i \le B_n\} $$ where for simplicity one can choose $A_n,\, B_n \in \mathbb{N}$ and $nB_n \ge A_n \ge B_n$.
The question is how to give a sharp estimation of cardinality of $F_{n}$? More specifically I want to choose $A_n,\, B_n \to +\infty$ is $o(n)$ i.e $\lim_{n\to+\infty}\frac{A_n + B_n}{n} = 0,\, \lim_{n\to+\infty}A_n = \lim_{n\to+\infty}B_n = +\infty$ and control the cardinality of $F_n$ as $$ \# F_n \le e^{o(n)},\quad i.e. \lim_{n\to+\infty}\frac{\ln \# F_n}{n} = 0 $$ Is it possible to achieve the cardinality control above? I am not familiar with combinatorics and I'll appreciate any help.
Unfortunately this cannot be done because $$ [\frac{A_n}{n},B_n]^{n} \subset F_n $$ so the cardinality has a lower bound $$ \# F_n \ge (B_n - \frac{A_n}{n})^{n} $$ and $\lim_{n\to+\infty}\frac{\ln \# F_n}{n} \ge \limsup_{n\to+\infty}\ln B_n = +\infty$.