The Schur positivity of Lucas binomial differences

Source abstract

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 $rs<uv$. Using Lucas atoms, we establish $e$-positivity for degree-adjusted Lucas multinomial differences under dominance order.