Exact Invertibility Probability for Binary Matrices with Two Ones in Every Row and Column

Let \(A_{N}\) be uniform among the \(N\times N\) zero–one matrices having two ones in every row and every column, with diagonal ones allowed. We prove an exact coefficient formula for \(\mathbb P(\det A_{N} \neq 0)\) for every \(N \ge 2\). If\(D(z)=e^{-z/2}(1-z)^{-1/2},\quad H(z)=e^{-z/2}(1+z)^{1/4}(1-z)^{-1/4},\)where the branches are defined by their power series at zero, then the probability is \([z^{N}] H(z) / [z^{N}] D(z)\). We also obtain the parity‑sensitive expansion\(\mathbb{P}\left(\det A_{N} \neq 0\right)=\frac{2^{1 / 4} \sqrt{\pi}}{\Gamma(1 / 4)} N^{-1 / 4}+(-1)^{N+1} \frac{e \sqrt{\pi}}{2^{9 / 4} \Gamma(3 / 4)}\left(N^{-3 / 4}+N^{-7 / 4}\right)+O\left(N^{-9 / 4}\right).\)The proof derives the conditioned Ewens representation from its matching fibres, proves the cycle determinant criterion, and retains the contribution of the second dominant singularity at \(z=-1\).