Dickson/determinant type polynomial (updated) For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}
consisting of all products of all non-zero linear forms $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero. 
Now let \begin{equation} \alpha_j = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation} 
and
\begin{equation} D_{k,\ell}= \det \begin{pmatrix} x_1^{\alpha_1} & \cdots & x_k^{\alpha_1} \\
\vdots & \vdots & \vdots \\
x_1^{\alpha_k} & \cdots & x_k^{\alpha_k} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k},  \end{equation} 
where $S_k$ is the symmetric group.
(Note that $\alpha_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.) 
Question: It is a classical fact that $D_{k,\ell}=P_{k,\ell}$ when $\ell =2, k$. Is it true that, for arbitrary $2\leq \ell \leq k$, $D_{k,\ell}$ is $P_{k,\ell}$ plus (possibly) some additional terms? 
 A: Is this true? Sage disagrees:
P.<a,b,c,d> = PolynomialRing(GF(2))
xs = P.gens()
M = Matrix(P, [[xs[i] ** (1 + binomial(j, 1) + binomial(j,2)) for j in range(4)] for i in range(4)])
M.determinant().factor()

returns
d * c * (c + d) * b * (b + d) * (b + c) * a * (a + d) * (a + c) * (a + b) * (a^3*b + a^2*b^2 + a*b^3 + a^3*c + b^3*c + a^2*c^2 + b^2*c^2 + a*c^3 + b*c^3 + a^3*d + b^3*d + a*b*c*d + c^3*d + a^2*d^2 + b^2*d^2 + c^2*d^2 + a*d^3 + b*d^3 + c*d^3)

This has all the right factors with $1$ or $2$ addends (which is no surprise), but the remaining degree-$4$ factor does not factor further as you want it to. For comparison, the product of the factors with $3$ addends should be
P.prod([P.sum(xs)-r for r in xs])

which is
a^3*b + a*b^3 + a^3*c + b^3*c + a*c^3 + b*c^3 + a^3*d + b^3*d + a*b*c*d + c^3*d + a*d^3 + b*d^3 + c*d^3

Generally, I think you can learn something about these polynomials from: I. G. Macdonald, Schur functions: Theme and variations, Séminaire Lotharingien de Combinatoire 1992, volume 28, page B28a (specifically, the 7th variation, which does not assume much familiarity with the preceding ones). But do not expect explicit formulas beyond the $\ell = 2$ and $\ell = k$ cases.
A: For a given $\sigma\in S_{k}$, the coefficient of $x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k}$ in $P_{k,l}$ is 1. So $D_{k,l}$ is $P_{k,l}$ plus (possibly) some additional terms.
