= Uniform layer of the Boolean cube
{title2=$\Omega_r=\{x\in\{0,1\}^n:\sum_i x_i=r\}$}
= Uniform layers of the Boolean cube
{synonym}
The uniform layer consists of the <characteristic vectors of sets> of size $r$ in $[n]$. It has size $\binom nr$, a <binomial coefficient>. Restricting <polynomials> to a uniform layer introduces identities absent on the entire <Boolean lattice>, since the coordinate sum is fixed. In particular, <homogenisation on a uniform layer> spans all restricted <multilinear polynomials> of degree at most $s\leq r$ using just the degree-$s$ <monomials>.
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