I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head.
(2.2.8) ~ Let $a(m,n)$ be the number of representations of $m$ as the sum of integers in two partitions into distinct parts for which the first partition must have exactly $n$ more elements than the second, and the second is allowed to have $0$s. If $n$ is negative, then the second parition has exactly $-n$ more elements than the first. […] Find a bijection between the partitions counted by $a(m,n)$ and those counted by $a(m-n,-n)$.
[emphasis added]
My best guess at what this is saying is that we are looking at pairs of partitions of $m$ with some properties, and the relation between them is that one has $n$ more parts than the other. But then the statement is not true:
- $a(4,1)=1$ because only $(\{3,1\}, \{4\})$ has both partitions with distinct parts such that the second contains more than the first.
- but $a(3,-1)>1$ because both $(\{3\}, \{3,0\})$ and $(\{3\}, \{2,1\})$ have the same properties.
That said, there is a lot of language here that I'm not used to.
- Does the word "representations" mean something special here?
- What word is "into" modifying?
- What is an "element" of a partition?
Probably this is really obvious, but right now I'm not seeing it. Any help is appreciated.