Unimodality of $q$-Fibonomial coefficients for small cases

TL;DR

This paper proves the unimodality of $q$-Fibonomial coefficients for small cases ($n\, extless= 3$), using combinatorial and algebraic methods, and discusses related conjectures.

Contribution

It confirms the unimodality conjecture for small $n$ and provides combinatorial and algebraic proofs, also extending to certain product cases.

Findings

Unimodality proven for $n\, extless= 3$ cases.

Combinatorial proof for $n=2$ case.

Algebraic proof established for all three cases.

Abstract

Bergeron--Ceballos--K\"ustner 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.