"${n}$-$1$" paths on lattice graphs. Random walks

In this paper, we consider graph-lattices with "$n$-$1$" constraints on attainability whose vertices are located at points of the plane with nonnegative integer coordinates. Two arcs emerge from each vertex: a horizontal arc comes to the nearest right vertex and a vertical arc to the nearest upper vertex. In the case of the "$n$-$1$" attainability, reachable paths are paths that satisfy the additional condition, namely, the multiplicity $n$ of the number of arcs in the maximal segments of paths consisting only of horizontal arcs. This restriction does not apply to the final segment of a path consisting of horizontal arcs. We obtain a formula for the number of "$n$-$1$" paths leading from a vertex to another vertex and also a formula for the number of such paths passing through a given vertex of the graph-lattice. Random walks along "$n$-$1$" paths on graph-lattices are considered. It is shown that such processes can be locally reduced to Markov processs on subgraphs determined by the type of the initial vertex. Also, we obtain formulas for the probabilities of transition from a vertex to another vertex along "$n$-$1$" paths and some combinatorial identities on Pascal's triangle.