FIG. 2.

Example of a single term contributing to the matrix element M_t at lowest order in time (t^N, here with N = 3). Here, all spins are flipped from down to up by a particular pairing off of the spins between the $\mathcal{A}$ and $\mathcal{B}$ sublattices. The depicted process contributes a term (J_1,2 × J_3,1 × J_2,3) × (t³/3!) to M_t. The set of all possible ways to pair the spins in sublattice $\mathcal{A}$ with the spins in sublattice $\mathcal{B}$ is in one-to-one correspondence with terms in the permanent of the matrix J_i,j and thus M_t is proportional to this permanent.