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For a given number $n >0$ is there a way to get combination that add up to this number??

for example :

if $n=6$ then numbers that add up are $5+1,4+2,3+2+1$ so the combination is 3

if $n=4$ then numbers that add up are $3+1$ the combination is 1

i.e each number that adds up is unique and the equation don't repeat i.e $3+1$ is same as $1+3$.

what is the formula to get the such combination ??

UPDATE ok after reading few articles and finally understanding how the partition number theory works , the final question i have is given the partition of distinct numbers (which is equal to odd partiotion)

$$ \sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}. $$

how can i get the coefficient of a variable say $x^5$ of $x^{34}$

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  • $\begingroup$ You look for partitions , but apparently for those having no duplicates in the summands. There should be a formula for the number of such partitions, but I only know it for the case that duplicates are allowed. $\endgroup$
    – Peter
    Commented Aug 27, 2020 at 11:39
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – JMoravitz
    Commented Aug 27, 2020 at 12:03

1 Answer 1

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There are many formulas to calculate the partition function $p(n)$ of an integer $n$. The last was found by Professor Ken Ono in 2011. More precisely, one has, $$p(n)=\frac{\text{Tr}^{(p)}(P;n)}{24n-1}$$ The number $p(n)$ are traces of the Poincare series $P(z)$ which is defined in Prof. Ken Ono's paper. For further info, consider the following article: https://uva.theopenscholar.com/files/ken-ono/files/097.pdf

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  • $\begingroup$ $Tr^(p)(P;n)$ i'm a bit confused here what does P represent ans what does $Tr$ expands to?? $\endgroup$
    – Akash Jain
    Commented Aug 27, 2020 at 12:56
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    $\begingroup$ @AkashJain $\text{Tr}$ is basically the trace. For definitions of $P(z)$ kindly refer to the article I mentioned in the answer. $\endgroup$
    – Anand
    Commented Aug 27, 2020 at 14:13
  • $\begingroup$ hey anand just updated my question , so basically i need to replicate this theorem in a programmitical code so the the question now is how to find a coefficent of a power of x in the partition theorem for distinct numbers $\endgroup$
    – Akash Jain
    Commented Aug 27, 2020 at 15:07
  • $\begingroup$ @AkashJain I don't feel your problem is suitable for MSE in that case. $\endgroup$
    – Anand
    Commented Aug 27, 2020 at 15:44
  • $\begingroup$ ok let me rephrase it ,i wish to calculate the coefficient of a value say $x^90$ how can i do it mathematically?? $\endgroup$
    – Akash Jain
    Commented Aug 27, 2020 at 16:17

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