Counting filter restricted paths in $\mathbb{Z}^2$ lattice

Abstract

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.