Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
$31555584/2^{13}$ = 3852. I think for $q = 3$ each perfect matching corresponds $2^{(1+3+3^2)}$ such polynomials because for every pair $U$, $V$ of $1$-dimensional subspaces with $f(U) = f(V)$ we can take any permutation of the non-zero elements of $V$ to define another permutation polynomial corresponding to the same perfect matching and since there are $13$ $1$-dimensional subspaces we have that factor. In general I'll have to be careful about this and I think I should re-formulate the conditions. (Thank you for pointing it out and I apologise for so many errors)
The number of perfect matchings in the incidence graph of $PG(2,3)$ is certainly $3852$. You can also check it here: oeis.org/A000794. So, there might be some error in the interpretation of a perfect matching as a permutation polynomial. I would check it again and see if I can find the error.
