The Schur positivity of Lucas binomial differences
For positive integers $a\le b\le c\le d$ with $ad=bc$, we prove that the Lucas binomial coefficients satisfy $(-1)^{a+1}(L_{b,c}[X]-L_{a,d}[X])\in\mathbb{N}[e_1[X],e_2[X],h_2[X]]$, resolving Bergeron's conjecture on Schur positivity in two variables. We also prove the same positivity for $L_{u,v}[X]-e_1[X]^{uv-rs}L_{r,s}[X]$ when $rs
Authors
- Dun Qiu (ORCID: https://orcid.org/0000-0002-0551-9472)
- Weichen Wang (ORCID: https://orcid.org/0000-0002-2896-2004)
- Lilan Dai
- Minhao Zhang
Publication Details
- Journal
- arXiv (Cornell University)
- Published
- 2026-10-26
- DOI
- https://doi.org/10.48550/arxiv.2610.35465
- Primary Topic
- Advanced Combinatorial Mathematics
- Type
- preprint