Let $A$ be an $n \times n$ real orthogonal matrix, $A^T A = I$. For subsets $S, S'$ of $\{1,\dots, n\}$, let $A_{S,S'}$ denote the restriction of $A$ to rows indexed by $S$ and columns indexed by $S'$. I'm wondering if there is a general formula or bound for (the absolute value of) $$\sum_{S' \subseteq\{1,\dots, n\}:|S'| = |S|} \text{det}(A_{S,S'})$$ for a given $S \subseteq \{1,\dots, n\}$?
For instance, in the special case where $A$ is not only orthogonal, but also a permutation matrix, this sum evaluates to $+1$ or $-1$, since for fixed $S$, there is exactly one $S'$ for which $A_{S,S'}$ has a nonzero entry in each row (and for that $S'$, $A_{S,S'}$ is itself a permutation matrix). But more generally, for orthogonal $A$, I'm interested in whether (the absolute value of) this sum can be bounded (as a function of $n$ and/or $|S|$). Is there a known reference for this problem? Even just knowing whether it's polynomially or exponentially increasing in $n$ (and/or $|S|$) would suffice as a starting point.