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$\def\char{\mathop{\rm char}}$In this paper by M. Budrevich and A. Guterman prove that for every $F_q$ with $\char F_q>2$ there are more matrices of order $n$ with zero determinant than those with zero permanent. In fact, their estimates can be strengthen to show the following. If $D(M_n(F_q))$ and $P(M_n(F_q))$ are the numbers of the corresponding matrices, then $$ D(M_n(F_q))-P(M_n(F_q))\geq (q-1)\prod_{k=3}^n(q^n-q^{k-1}). $$ (This is shown in Budrevich's dissertation.)