While trying to prove a result in group theory I came up with the following conjecture on partitions:
Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for all $i,j\in\mathbb{N}$. Suppose for $m\in\mathbb{N}$, $a(m)$ denotes the number of partitions of $m$ with each part at least $2$ . Then the following holds: $$\sum_{i+j=m} b(i,j)=a(m).$$
I need to prove or disprove the above conjecture. I checked first few cases where it holds true. I tried induction and some bijection arguments but did not succeed. Any idea will be appreciated.
You can give a bijection argument: Interpret $b(i,j)$ as the number of partitions of $m=i+j$ with greatest part $j$ occuring at least twice. Then the LHS counts the number of partitions of $m$ with greatest part occuring at least twice.
Now give a bijection of the set of partitions of $m$ with greatest part occuring at least twice to the set of partitions of $m$ with each part at least $2$ by rotation of the corresponding Ferrers diagram.