I'm looking for operations on binary numbers that have the same number of the available digits, $0$ and $1$, such as $100011$, $100101$ and $10$. These could be integers as the examples given are, or restricted, for example to specific rational ranges.
That preserve this quality, so that the $1$'s count is still equal to the $0$'s count.
Can an example be given?
I'm interested in this for possible compute applications, however this question as posed seems a better fit for maths.SE.