Below I consider vectors of same length $n$ and consisting of only $1$ and $-1$ elements. Let's call vector $v_1$ dominating for $v_2$ if count of $-1$ in $v_1$ is not less then in $v_2$ and position of $i$-th $-1$ in $v_1$ is less or equal then position of $i$-th $-1$ in $v_2$ for all $i$. I have many similar almost symmetric sets of dominating vectors, like this:
(+ + - - - + - + - + - - - + + +)
(+ - + - - + - + - + - - + - + +)
(+ - - + - + - + - + - + - - + +)
(- + + - - + - + - + - - + + - +)
(- + - + - + - + - + - + - + - +)
(- - + + - + - + - + - + + - - +)
(+ + + - - + - + - + - - + + + -)
(+ + - + - + - + - + - + - + + -)
(+ - + + - + - + - + - + + - + -)
(- + + + - + - + - + - + + + - -)
Is there relatively simple way to find formula for number of (union of) vectors which are dominated by these vectors?
This problem arises from problem of counting hyperplanes crossing chamber of some arrangement. I don't tell details because arrangement is too specific.