Let $n$ denote a positive integer, and consider the set $S$ of $n\times n$ symmetric matrices with zero diagonal (denoted $A$) so that for each $j$, $a_{ij}$ ($i=1,2,\dots,n)$ is a permutation of $0,1,\dots,n-1$. What is $S$'s cardinality?
Some ideas: we may let the symmetric group $S_n$ act on $S$ by conjugating by the permutation matrices; we may replace $1,\dots,n-1$ with variables $x_1,\dots,x_{n-1}$ [or with some values, I had in mind the $n-1$th roots of unity] and try and find the eigenvalues of $A\in S$. Any idea will be appreciated, thanks.
(trivial observation: $n$ has to be even. indeed, assume $n=2m+1$. since $a_{ij}=a_{kl}\neq 0$ implies $i,k,j,l$ all different (assuming $\{i,j\}\neq \{k,l\}$), there are at most $2m$ elements in $A$ equal to $r$, for any $r=1,\dots,n-1$, even though there need to be $n$ such elements.)