  Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson,
and the second author have conjectured that the coefficient of any Schur
function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in
$\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case
$k=2$; We show that the coefficient of any Schur function $s_\lambda[X]$ in
$\Delta_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.
