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So I am supposed to show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of n for which no part is divisible by 3. and there is one part which I cannot easily show mathematically (That part was found in the penultimate answer to Partitions of $n$ into distinct odd and even parts proof). As said I understand the proof but there is this small algebraic expression I don´t understand, how they are equal.

$$\prod_{k\ge 0} \frac{1}{1-z^{3k+1}} \prod_{k\ge 0} \frac{1}{1-z^{3k+2}}=$$ $$=\prod_{k\ge 1} \frac{1-z^{3k}}{1-z^k}$$

I would appreciate any kind of help?

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  • $\begingroup$ Multiply both sides with $\prod_{k \geqslant 1} \frac{1}{1 - z^{3k}}$. $\endgroup$ – Daniel Fischer May 17 '20 at 16:03
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If in $$\prod_{k\ge 1}{1-z^k} $$ you split $k$ as $3j, \; 3j+1,\; 3j+2$, you can easily arrive to check the identity

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