Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Subtracting $\lceil n^\beta\rceil-1$ from every $x_i$, translates the problem translates to the number of partitions of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$
Subtracting $\lceil n^\beta\rceil-1$ from every $x_i$, the problem translates to the number of partitions of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$
Subtracting $\lceil n^\beta\rceil-1$ from every $x_i$ translates the problem to the number of partitions of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$
Subtracting $\lceil n^\beta\rceil-1$ from every $x_i$, the problem translates to the number of partitions of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$
