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The Schur positivity of Lucas binomial differences
Abstract
For positive integers a≤ b≤ c≤ d with ad=bc, we prove that the Lucas binomial coefficients satisfy (−1)a+1(Lb,c[X]−La,d[X])∈ℕ[e1[X],e2[X],h2[X]], resolving Bergeron's conjecture on Schur positivity in two variables. We also prove the same positivity for Lu,v[X]−e1[X]uv−rsLr,s[X] when rs<uv. Using Lucas atoms, we establish e-positivity for degree-adjusted Lucas multinomial differences under dominance order.
Related
BibTeX
@misc{dai2026schur,
title = {{The Schur positivity of Lucas binomial differences}},
author = {Lilan Dai and Dun Qiu and Weichen Wang and Minhao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35465},
eprint = {2610.35465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35465},
}
RIS
TY - UNPB
TI - The Schur positivity of Lucas binomial differences
AU - Lilan Dai
AU - Dun Qiu
AU - Weichen Wang
AU - Minhao Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35465
UR - https://arxiv.org/abs/2610.35465
AB - For positive integers a≤ b≤ c≤ d with ad=bc, we prove that the Lucas binomial coefficients satisfy (−1)^a+1(L_b,c[X]−L_a,d[X])∈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−rsL_r,s[X] when rs<uv. Using Lucas atoms, we establish e-positivity for degree-adjusted Lucas multinomial differences under dominance order.
ER -