Unimodality of $q$-Fibonomial coefficients for small cases

Abstract

Bergeron--Ceballos--Küstner introduced the $q$-Fibonomial coefficients \( \qfibonom{m+n}{n}\), gave a combinatorial interpretation of the $q$-Fibonomial coefficients via a weighted path-domino tiling model, and conjectured that these polynomials are unimodal. We prove the conjecture for $n\leq3$. For the $n=2$ case, we give a combinatorial proof of both unimodality and symmetry by defining a nearly symmetric saturated chain decomposition on the set of tilings. For all three cases, we give an algebraic proof. Finally, for the $n=3$ case, we establish a more general unimodality result for certain products of $q$-analogs and propose several related conjectures.