Let $k$ be an integer and $[k] = \{1,2,...,k\}$. Let $\mathcal{A}$ be the set of tuples $(I_1, I_2, ..., I_{k+1})$ such that for all $j$, $I_j \subseteq [k]$ and $|I_j|$ is even. Also, let $\mathcal{A}_i$ be the subset of $\mathcal{A}$ such that $|I_1| + |I_2| + ... + |I_{k+1}| = i$. In perticular, $\mathcal{A}_i = \emptyset$ if $i$ is odd.
Let $\mathcal{B}$ be defined the same way, but interchanging the roles of $k$ and $k+1$. So, it is the set of tuples $(I_1, I_2, ..., I_{k})$ such that for all $j$, $I_j \subseteq [k+1]$ and $|I_j|$ is even. Define $\mathcal{B}_i$ analogously.
I want to show that for all $i \leq k^2-1$, $i$ even, there is an injection $\mathcal{A}_i \rightarrow \mathcal{B}_i$. I have been able to show that $|\mathcal{A}_i| \leq |\mathcal{B}_i|$, but I'm looking for a combinatorial proof, if one exists.
What I tried so far
One obvious guess would be to send $(I_1, I_2, ..., I_{k+1})$ to $(I_1', I_2', ..., I_k')$, where $I_j' = I_j \cup \{k+1\}$ if $j \in I_{k+1}$ and $I_j' = I_j$ else. But then, some of the subsets would not be of even size.
To correct it, we can use the bijection $I \mapsto I \triangle \{1\}$ between subsets of odd size and those of even size, where $\triangle$ is the symmetric difference. So, we add $1$ to $I$ if $1 \not\in I$, and substract $1$ from $I$ if $1 \in I$. But then, the total sum of the sizes of $(I_1, I_2, ..., I_{k+1})$ is changed.
Does anyone know of an injection that would both preserve the even size of all subsets and the total size of the subsets in the tuple?
Example
Let $k=3$. An element of $\mathcal{A}_6$ is
$$ (\{1,2\}, \emptyset, \{2,3\} , \{2,3\}) $$
Using what I tried so far, the first injection would send it to
$$ (\{1,2\}, \{4\}, \{2,3,4\}) $$.
Using the symmetric difference with $\{1\}$ on subsets of odd size (those who contain $4$), we obtain
$$ (\{1,2\}, \{1,4\}, \{1,2,3,4\}) $$
But then, this tuple is in $\mathcal{B}_8$, and not in $\mathcal{B}_6$, which is what I would like.
Is there a way to construct such an injection that preserves the sum of the sizes of the subsets?