<s>[INST] Parity of Partition Function Let $T(n)$ denote the number of partitions of $n$ into parts not congruent to $3$ mod $6$.  Deduce that $T(n)$ is also the number of partitions of $n $ in which odd parts appear at most twice (even parts appear without restriction).  Can you give a combinatorial proof that $T(n)$ is even whenever $n$ is congruent to $2$ or $3$ mod $4$?
 [/INST] For the first part, note that the generating function for partitions with parts not congruent to three mod 6 is
$$\prod_{k\ge 1} \frac{1}{1-z^k} \prod_{k \ge 1} (1-z^{6k-3})
= \prod_{k\ge 1} \frac{1}{1-z^{2k}} \prod_{k\ge 1} \frac{1}{1-z^{2k-1}}
 \prod_{k \ge 1} (1-z^{6k-3})
\\ =  \prod_{k\ge 1} \frac{1}{1-z^{2k}}
\prod_{k\ge 1} \frac{1-z^{6k-3}}{1-z^{2k-1}}
=   \prod_{k\ge 1} \frac{1}{1-z^{2k}}  
\prod_{k\ge 1} \left(1+z^{2k-1}+z^{4k-2}\right).$$
This concludes  the first part of the exercise. 
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