{"paper":{"title":"Unimodality of $q$-Fibonomial coefficients for small cases","license":"http://creativecommons.org/licenses/by/4.0/","headline":"q-Fibonomial coefficients are unimodal for all n at most 3","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan B. Connelly, Ezekiel Ito, Kacey Yang, Olha Shevchenko, Thomas C. Martinez","submitted_at":"2026-05-12T23:44:53Z","abstract_excerpt":"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 cert"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the conjecture for n≤3. 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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The weighted path-domino tiling model introduced by Bergeron--Ceballos--Küstner correctly interprets the q-Fibonomial coefficients and that the algebraic identities for q-analogs hold without additional restrictions for the small n considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"q-Fibonomial coefficients are unimodal for n≤3, with a combinatorial proof of unimodality and symmetry for n=2 via nearly symmetric saturated chain decompositions on tilings and algebraic proofs for n≤3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"q-Fibonomial coefficients are unimodal for all n at most 3","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1c49ebf161dba205ad0285783d1a171cb3d60d538f944edc324a5145a0b49521"},"source":{"id":"2605.12822","kind":"arxiv","version":1},"verdict":{"id":"8f5a7bb6-e5a9-4cb3-874a-2d1decad5049","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:21:27.682099Z","strongest_claim":"We prove the conjecture for n≤3. 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.","one_line_summary":"q-Fibonomial coefficients are unimodal for n≤3, with a combinatorial proof of unimodality and symmetry for n=2 via nearly symmetric saturated chain decompositions on tilings and algebraic proofs for n≤3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The weighted path-domino tiling model introduced by Bergeron--Ceballos--Küstner correctly interprets the q-Fibonomial coefficients and that the algebraic identities for q-analogs hold without additional restrictions for the small n considered.","pith_extraction_headline":"q-Fibonomial coefficients are unimodal for all n at most 3"},"references":{"count":19,"sample":[{"doi":"","year":null,"title":"Symmetry, Integrability and Geometry: Methods and Applications , language =","work_id":"e6d816c8-66ed-49b6-835b-a2bb2b2f3052","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Benjamin, Arthur and Quinn, Jennifer J. , address =","work_id":"fa8b6523-cb71-4836-88a7-cacad4c84768","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"1989 , publisher =","work_id":"cbe25d5b-6871-4096-81ac-d3dc98561d85","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"1878 , publisher =","work_id":"8d0cca3b-9dec-4795-b33a-a6fe6bfa357e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"O'Hara, Kathleen M. , journal =. 1990 , publisher =","work_id":"ad6ba07b-fb85-41a9-b018-d314a51c7841","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"ae120834c98d568633a04da18f251e69c7934efbd13aa69b39e46fedb33fbe80","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}