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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:

1. 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

Are any of these acceptable?

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:

1. 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

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:

1. Question on permutation depths. The answer to this questions suggests n-tuples as the correct name.

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:

1. 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

Are any of these acceptable?

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?

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:

1. Question on permutation depths. The answer to this questions suggests n-tuples as the correct name.