It is shown that the hopping of a single excitation on certain
triangular spin lattices with non-uniform couplings and local magnetic
fields can be described as the projections of quantum walks on graphs of
the ordered Hamming scheme of depth 2. For some values of the parameters
the models exhibit perfect state transfer between two summits of the
lattice. Fractional revival is also observed in some instances. The
bivariate Krawtchouk polynomials of the Tratnik type that form the
eigenvalue matrices of the ordered Hamming scheme of depth 2 give the
overlaps between the energy eigenstates and the occupational basis
vectors.