= Maximum-support generator fibre
{title2=$|\mathcal F_G|=\binom{n-m}{k-|G|}$}
Suppose a <left-compressed set family> of $k$-sets has an <antichain> of generators on $[m]$, closed under left shifts followed by taking inclusion-minimal members. For a generator $G$ containing $m$, the members generated only by $G$ have intersection with $[m]$ exactly $G$. If another prefix coordinate $i$ occurred, the left shift $G-\{m\}+\{i\}$ would supply another generator. Conversely, an exact prefix $G$ contains no other generator by the <antichain> condition. The free tail gives the displayed <binomial coefficient>.
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