  We derive a path counting formula for two-dimensional lattice path model on a
plane with filter restrictions. A filter is a line that restricts the path
passing it to one of possible directions. Moreover, each path that touches this
line is assigned a special weight. The periodic filter restrictions are
motivated by the problem of tensor power decomposition for representations of
quantum $\mathfrak{sl}_2$ at roots of unity. Our main result is the explicit
formula for the weighted number of paths from the origin to a fixed point
between two filters in this model.
