I've read in several sources that the number of facets of the Birkhoff polytope $\mathcal{B}(n)$ is $n^2$.
Is this supposed to hold when $n=2$? Since $\mathcal{B}(2)$ has dimension $1$, the facets would be the two $0$-dimensional vertices, which are the two permutation matrices below: $$\begin{pmatrix} 1 & 0 \\ 0 &1 \end{pmatrix} \text{ and } \begin{pmatrix} 0 & 1 \\ 1 &0 \end{pmatrix}$$ However, the claim is that there should be $2^2 = 4$ facets. None of my sources have given any restriction on $n$. What am I missing?