$k$-homogeneous Latin Squares, their transversals and condition of pseudo-orthogonality

We consider $k$-quasi-group operations defined on a ring of integers using addition and permutations that have finite sets of mobile elements contained among the numbers $0,1,\ldots,m-1$. The question of the article is number of coincidences of the values of these operations on all possible sets of length $k$ composed of these numbers is studied. For two affine permutations on these numbers, formulas for the number of coincidences and the corresponding generating functions are obtained.