Given a field $k$ of characteristic 0, one can show that the space of translation invariant polynomials in $n$ variables, homogeneous of degree $N$ is spanned by
$$ \{ (x_1-x_2)^{d_1}(x_2-x_3)^{d_3}\ldots (x_{n-1}-x_n)^{d_{n-1}} \ |\ d_1+\cdots + d_{n-1}=N\} $$
If we consider the span of elements of this form, but allow the signs to vary, i.e. $\prod (x_i \pm x_{i+1}) $, we obtain a significantly larger space, spanned by polynomials that can somehow be considered 'pseudo translation invariant'. However, this spanning set is not linearly independent
$$ (x_1 + x_2) - (x_2 - x_3) = (x_1+x_3) $$
Can we provide a better bound on the dimension of this space than the $2^{n-1}\binom{N+n-2}{n-2} $ obtained by counting the number of spanning elements?