Towards counting paths in lattice path models with filter restrictions and long steps

In this paper we introduce the notion of congruence for regions in lattice path models. This turns out to be useful for deriving path counting formula for the auxiliary lattice path model in the presence of long steps, source and target points of which are situated near the filter restrictions. This problem was motivated by the fact, that weighted numbers of paths in such model mimic multiplicities in tensor power decomposition of $U_q(sl_2)$-module $T(1)^{\otimes N}$ at roots of unity. We expand on combinatorial properties of such model and introduce the punchline of a proof for explicit path counting formula.