A Combinatorial Interpretation of the Lucas-Nomial Coefficients in Terms of Tiling of Rectangular Boxes

Generalized binomial coefficients are considered. The aim of this paper is to provide a new general combinatorial interpretation of the Lucas-nomial and \( (p,q) \)-nomial coefficients in terms of tiling of \( d \)-dimensional rectangular boxes. The recurrence relation of these numbers is proved in a combinatorial way. To this end, our results are extended to the case of corresponding multi-nomial coefficients.