Generalization of Fibonomial Coefficients

Following Lucas and then other Fibonacci people Kwaśniewski had introduced and had started ten years ago the open investigation of the overall F -nomial coefficients which encompass among others Binomial, Gaussian and Fibonomial coefficients with a new unified combinatorial interpretation expressed in terms of cobweb posets’ partitions and tilings of discrete hyperboxes. In this paper, we deal with special subfamily of T -nomial coefficients. The main aim of this note is to develop the theory of T -nomial coefficients with the help of generating functions. The binomial-like theorem for T -nomials is delivered here and some consequences of it are drawn. A new combinatorial interpretation of T -nomial coefficients is provided and compared with the Konvalina way of objects’ selections from weighted boxes. A brief summary of already known properties of F -nomial coefficients is served.