# Definition

Call a partition $\lambda$ of an even integer $2n$ "black-white balanced" if the following equivalent conditions are satisfied:

- In the usual (Ferrers-)Young diagram of $\lambda$, if you color alternating squares black and white in a checkerboard pattern, there are an equal number of black and white squares.
- The Young diagram of $\lambda$ can be covered by dominoes.
- The partition $\lambda$ has empty 2-core.
- If you sort the parts of $\lambda$ (as is usual), then the number of odd parts in even places is equal to the number of odd parts in odd places. (A part $\lambda_i$ is odd if the number $\lambda_i$ itself is odd; the place is odd or even based on the parity of the index $i$.)
- If you sort the parts of $\lambda$, then $\sum_i (-1)^i(1-(-1)^{\lambda_i}) = 0$.

The number of black-white balanced partitions of $2n$ is OEIS sequence A000712, which begins $1$, $2$, $5$, $10$, $20$, $36$, $65$, $110$. The name of this sequence is "Number of partitions of n into parts of 2 kinds". Alternatively, it is the convolution of the partition numbers with themselves. The description above appears on the linked page as "Also equals number of partitions of 2n in which the odd parts appear as many times in even as in odd positions."

# Question

In other words, there appears to be a bijection between black-white balanced partitions $\lambda$ of $2n$ and ordered pairs of partitions ($\lambda_1$, $\lambda_2$) of a *combined* $n$ (that is, if $\lambda_1$ is a partition of $n_1$ and $\lambda_2$ is a partition of $n_2$, then $n=n_1+n_2$).

Can anyone describe this bijection, or provide a reference for it?

## Remarks

I'm fairly sure I'm just a Google search away from finding a reference, but I just haven't succeeded yet. I found this question on Mathematics Stack Exchange and this identical question on MathOverflow asking about a bijection between *strict* (ie, distinct parts) black-white balanced partitions of $2n$ and unrestricted partitions of $n$ (not pairs thereof). The paper "Balanced partitions" by Sam Vandervelde also proves this result.

uniquedomino tiling istwicethe number of unrestricted partitions of $n$. It sketches a proof using abacus diagrams. Armed with this search term, I could find lots of interesting stuff to read, but still no reference for my desired result. $\endgroup$Enumerative Combinatorics, vol. 2. The proof is a disguised version of the abaci proof of James and Kerber. Some additional references are in the solution to the exercise. $\endgroup$1more comment