  We consider a variant of Glauber dynamics of Ising spins on a one-dimensional
lattice, where each spin flips according to the relative state of the spin to
its left. Moreover, each bond allows for two rates; flips which equalize
nearest neighbor spins, and flips which "unequalize" them. In addition, the
leftmost spin flips depending on the spin at that site. We explicitly calculate
all eigenvalues of the transition matrix for all system sizes and conjecture a
formula for the normalization factor of the model. We then analyze two limits
of this model, which are analogous to ferromagnetic and antiferromagnetic
behavior in the Ising model for which we are able to prove an analogous formula
for the normalization factor.
