Combinatorial Interpretations of $q$-Fibonacci Numbers and Their Binomial Analogues (2305.01838v2)

Abstract: The Fibonomial coefficients are well-known analogues of the classical binomial coefficients. In 2009, Sagan and Savage introduced a combinatorial interpretation for these coefficients, based on tiling a rectangular grid. More recently, Bergeron extended this work by providing a similar interpretation for the q-Fibonomial coefficients, using weighted tilings of a rectangular grid. Inspired by Bennett's model, Bergeron also developed a staircase tiling model for the q-Fibonomial coefficients. While Bergeron's proofs for the rectangular grid model relied on induction, and the staircase model on bijective correspondences with the rectangular grid model, these approaches lacked deeper structural insights. In this paper, we propose a novel model for the q-Fibonacci numbers that generalizes Bergeron's approach. This new model not only enables us to prove several identities related to q-Fibonacci numbers but also provides a non-bijective proof for the staircase model of the q-Fibonomial coefficients, offering greater structural clarity. Additionally, we demonstrate new identities involving the q-Fibonomial coefficients using this refined rectangular grid model, further enhancing the combinatorial understanding of these mathematical objects.