Matrix approach to the problem of matrix partitioning

We derive upper and lower bounds on the number of all variants a rectangular M×N matrix can be partitioned into fragments. Next the problem of matrix partitioning is considered as a particular example of a more general problem of constructing two-dimensional Markov processes (fields) on discrete rectangular lattices. We discuss a matrix-theoretical approach to the problem to explore the structure of discrete fields defined by a given matrix of local interaction