We may treat a given permutation as a vector variable, so that permutation s$_n$ is a variable that takes values among all possible number sequences of length $n$, which have elements in the rank order indicated by s$_n$. For example, if s$_3 = (1, 3, 2)$, then possible values of s$_3$ include $(.5, 83, \pi), (\sqrt2, 3, 2)$, etc. The solution to an equation with a permutation variable is a possibly-infinite set of $n$-length sequences. A system of simultaneous equations with permutation variables may be seen as a list of constraints on the solution set.
Let $(X, Y)$ be a bivariate sequence of $n$ real numbers $(x_i, y_i), i = 1, 2, ... , n$. Let $R$ be sequence of $k$ real numbers $r_j$, $j = 1, 2, ... , k$, with values in $[0, 2\pi)$. Let S$_{n,k}$ be a sequence of $k$ permutations, s$_{n,j}$.
Let $f(X, Y, r_j)$ be a function that does the following. 1) Rotates $Y$ by $X$ to $r_j$ radians, to obtain $(X', Y')$; 2) sorts $Y'$ by $X'$; and 3) returns the ascending rank order of the $n$ elements $y'_i$ in $Y'$, as permutation $s_{n,j}$.
Let $F$ be a system of $k$ simultaneous equations, each of the form $f_j(X, Y, r_j) = s_{n,j}$, with only $Y$ unknown. The solution to $F$ is a set of real-valued number sequences of length $n$ (i.e., all possible $Y$'s given known $X$, $R$, and S$_{n,k}$). The solution set is infinite, although the cardinality of the set obviously decreases with increasing $k$ and is 1 when $k$ is infinite.
I have three questions. First, is there a name for problems of this type: solving systems of simultaneous equations where one variable is a permutation indicating the rank order of elements in the solution? Note that this type of problem need not involve a rotation function. While I made $f$ a rotation in $r$ radians, I could have alternatively specified $f$ as, say, a polynomial in $X$ and $Y$ with $r_j$ a list of coefficients, or any similar function that returns a sequence (permutation) of $n$ rankings.
Second, given a real-valued number sequence of length $n$, what is the most efficient method for determining whether that sequence is a member of the solution set?
Third, is there an efficient method for characterizing the solution set? I intentionally leave "characterizing" vague, but it would include an indication of cardinality.