Generalization of Fibonomial Coefficients

TL;DR

This paper develops the theory of T-nomial coefficients, providing generating functions, a binomial-like theorem, and new combinatorial interpretations, extending the framework of Fibonomial and related coefficients.

Contribution

It introduces a new theory of T-nomial coefficients with generating functions, binomial-like theorem, and combinatorial interpretations, expanding the understanding of F-nomial coefficients.

Findings

Derived the generating functions for T-nomial coefficients

Established a binomial-like theorem for T-nomials

Provided a new combinatorial interpretation of T-nomial coefficients

Abstract

Following Lucas and then other Fibonacci people Kwasniewski 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 T-nomial coefficients is served.