If $0\leq\gamma<\alpha<1$ and $t=\lceil n^\gamma\rceil$ hold then how many positive solutions to the linear diophantine equation


$$x\_1+\dots+x\_t=\lceil n^\alpha\rceil$$


have the property


$$n^\beta\leq x\_1\leq x\_2\leq\dots\leq x\_t\leq\lceil n^\alpha\rceil$$ when $0\leq\beta<\alpha-\gamma$?



Number of positive integer solutions with a lower bound
Subtracting $\lceil n^\beta\rceil-1$ from every $x\_i$ translates the problem to the [number of partitions](https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_part_size_or_number_of_parts) of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p\_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$


