3
$\begingroup$

How I would write the generating function for a partition of a positive integer n with an even number of odd parts?

Any hints or suggestions will be greatly appreciated.

  • 0
    Let $P_k^{\,\mathrm{odd}}(n)$ be the set of partitions $\lambda\vdash n$ with $k$ odd parts, and $P_k(n)$ the set of partitions $\mu\vdash n$ with $k$ generic parts. Then we construct a bijection $f:P_{2k}^{\,\mathrm{odd}}(2n)\to P_{2k}(n+k)$ as follows: for each part of the partition $\lambda\in P_{2k}^{\,\mathrm{odd}}(2n)$, add $1$ and then divide by $2$. Hence $$\# P_{\mathrm{even}}^{\,\mathrm{odd}}(n)=\begin{cases}\sum_{k=1}^{n/2}\#P_{2k}\left( \frac{n}{2} +k\right) & \text{if }n\text{ is even,} \\ 0& \text{ if }n\text{ is odd}.\end{cases}$$ Just thought I'd put that out there.2012-01-29

1 Answers 1