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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 $0\leq\beta<\alpha-\gamma$?

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    $\begingroup$ Relevant: mathoverflow.net/questions/136726/… $\endgroup$
    – VS.
    Commented Nov 27, 2019 at 13:00
  • $\begingroup$ Is $\ell$ fixed? $\endgroup$
    – Anthony Quas
    Commented Nov 27, 2019 at 15:48
  • $\begingroup$ You mean $t$? $t$ can vary but even for fixed I am not sure of the behavior. $\endgroup$
    – VS.
    Commented Nov 27, 2019 at 16:56

1 Answer 1

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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: $$p_t(\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)).$$

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  • $\begingroup$ This is the case for fixed $t$... $\endgroup$
    – Anthony Quas
    Commented Nov 27, 2019 at 19:37
  • $\begingroup$ @AnthonyQuas: It is fixed as OP states that $t=\lceil n^\gamma\rceil$ for some $\gamma$. But even if it's variable, then summation over suitable $t$ will give the answer. $\endgroup$
    – Max Alekseyev
    Commented Nov 27, 2019 at 19:46
  • $\begingroup$ Sorry. Missed that $\endgroup$
    – Anthony Quas
    Commented Nov 27, 2019 at 20:20
  • $\begingroup$ @VS.: Please see the provided Wikipedia link for notation and asymptotic. $\endgroup$
    – Max Alekseyev
    Commented Nov 27, 2019 at 20:45
  • $\begingroup$ @MaxAlekseyev Is there expressions and asymptotics for $p_t$? $\endgroup$
    – VS.
    Commented Nov 27, 2019 at 21:39

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