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arXiv · 2605.12822

Unimodality of $q$-Fibonomial coefficients for small cases

Brendan B. Connelly·Ezekiel Ito·Thomas C. Martinez·Olha Shevchenko·Kacey Yang
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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.

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Brendan B. Connelly, Ezekiel Ito, Thomas C. Martinez, Olha Shevchenko, Kacey Yang. 2026-05-12. Unimodality of $q$-Fibonomial coefficients for small cases. https://arxiv.org/abs/2605.12822

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