We are interested in ReH-matrices because they help us to solve a complex operations research problem. However, these matrices themselves represent an interesting combinatorial structure. ReH-matrices can be initially computed by means of a simple enumeration (however a laborious method). Unfortunately, no formulas for explicit computation are known for most elements of ReH-matrices. We introduce ”perturbed permutations”. They can be used to compute the elements of ReH-matrices more effectively than enumeration. Limits of ReH-matrices are also significant. So-called Poisson equations are important for applications. If their solutions are ”monotone” then ReH-matrices correspond to optimal solutions of the above mentioned operations research problem. We prove that for a subset of such equations.