Graf-lattice has vertices at points with non-negative integer coordinates. From each vertex has two arcs: horizontal and vertical neighboring vertices (right and top). The transition probability for each of the arcs is equal to ½. Consider the problem of random walks on the vertices of the graph, without limitation on the achievable and with two types of limitations on achievable - mixed and magnetic. There is some combinatorial identities containing binomial coefficients from different layers of Pascal's triangle.

Keywords: directed graph, random walks, the probability of transition, achievable tops, Pascal's triangle, combinatorial identity