Suppose $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a partition of $2n$ where $n\in\mathbb N$ satisfying the following conditions:
(1) $\lambda_k=1$.
(2) $\lambda_i−\lambda_{i+1}\leq 1$ for every $i \leq k−1$.
(3) In the partition $\lambda$, the number of odd parts in odd places & the number of odd parts in even places are equal.
Here a part $\lambda_i$ is said to be in even place if $i$ is even, whereas $\lambda_i$ is said to be in odd place if $i$ is odd. $\lambda_i$ 's are called parts of $\lambda$ and $\lambda_i$ is called an odd part if it is odd & is called even part if it is even.
Now the question is to give a bijection between number of partitions of $2n$ satisfying the above conditions and number of partitions of $n$.