We derive a path counting formula for two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. The concept of congruence of connected regions in models of paths on a lattice is introduced. It turns out to be useful for deriving explicit formulas for calculating paths in an auxiliary path model in the presence of long steps, the beginning and end of which lie in filters. The problem is motivated by the fact that the weighted numbers of paths of such a model reproduce the multiplicities in the expansion of the tensor power of the two-dimensional U_q (sl_2)-module in roots of unity. The combinatorial properties of this model were studied, and a plan for the proof of the derivation of explicit formulas for calculating paths was outlined.
D.P. Soloviev. Towards counting paths in lattice path models with filter restrictions and long steps
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English (United Kingdom) 
Russian (Russia)