Can sample rank be defined as a function of a sample?
Suppose $X_1, X_2, \dots X_n \sim F$ where $F$ is defined on space $\mathcal{X}$. Consider the order statistics $X_{(1)}, X_{(2)}, \dots, X_{(n)}$. Then we know that rank of $X_i$ is $R_i$ if $X_i = X_{(R_i)}$. (I am assuming data are all different).
Is there any way to write rank as a function $f: \mathcal{X}^n \to \{1, \dots, n\}$?
Not sure if this helps, but assuming all data points are unique, then we can define the functions recursively
$$ R_{1} = \min \Big\{ j \in \{1,\dots,n\} : X_j = \min (X_1, \dots, X_n) \Big\} $$
$$R_{2} = \min \Big\{ j \in \{1,\dots,n \} : X_j = \min (X_{(-1)}) \Big\}$$ $$ \dots $$ $$R_i = \min \Big\{ j \in \{1,\dots,n \} : X_j = \min (X_{(-1,-2,\dots,-(i-1))}) \Big\}$$
Where $X_{(-1,-2,\dots,-(i-1))}$ is the datavector after removing $X_{(1)},\dots,X_{(i-1)}$, found using the prior defined rank functions for $R_1, \dots, R_{(i-1)}$.
While not exactly closed form, these functions are easily written in a program like R, if that's what your interested in doing.
Otherwise, more generally, we could write
$$R_i = \min \{j \in \{1,\dots,n\}: |\{k: X_k \leq X_j \}|=i \}$$
where $|\cdot |$ is set cardinality.