I'm creating two types of binary vector. The vectors have $M$ components and either $N$ ones or $(M-N)$ ones.
If you were to take all permutations of these vector types there would be a symmetry in the sets. One set could become the other by swapping the ones for zeroes and the zeroes for ones. For example, you could have the pair:
$$\begin{align}v &= [1, 0, 0, 1, 0, 1, 1] \\[10pt]v' &= [0, 1, 1, 0, 1, 0, 0]\end{align}$$
It feels like there should be a name for this type of a symmetry, and a name for the number of ones or conversely zeroes in the vector. But I can't think of what to search to find out what that name is. For the moment I'm using depth, but that doesn't seem right.
Is there a name for the number of ones in a binary vector, and a name for the symmetry between it and its logical inverse?
Potentially related:
- Question on permutation depths. The answer to this questions suggests n-tuples as the correct name.
Edit:
Suggestion:
As there doesn't appear to be a term that fully captures this idea, I suggest one of the following:
- Symmetric combination depth,
- Symmetric permutation depth,
- Symmetric depth,
- Symmetric hamming weight,
- Symmetric cardinality