Unimodality of $q$-Fibonomial coefficients for small cases
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910214307971072 |
|---|---|
| author | Connelly, Brendan B. Ito, Ezekiel Martinez, Thomas C. Shevchenko, Olha Yang, Kacey |
| author_facet | Connelly, Brendan B. Ito, Ezekiel Martinez, Thomas C. Shevchenko, Olha Yang, Kacey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unimodality of $q$-Fibonomial coefficients for small cases Connelly, Brendan B. Ito, Ezekiel Martinez, Thomas C. Shevchenko, Olha Yang, Kacey Combinatorics 11B39, 05A30, 05A10 F.2.2 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. |
| title | Unimodality of $q$-Fibonomial coefficients for small cases |
| topic | Combinatorics 11B39, 05A30, 05A10 F.2.2 |
| url | https://arxiv.org/abs/2605.12822 |