Edge boundary of a down-set in a cube
= Edge boundary of a down-set in a cube
{title2=$b_e(D)=n|D|-2\sum_{A\in D}|A|$}
For a <down-set> $D$ in the <Boolean lattice>, every member $A$ has all its $|A|$ immediate lower neighbors in $D$. Counting an internal edge at its upper endpoint gives $e(D)=\sum_{A\in D}|A|$. The degree identity for the <edge boundary> of the <hypercube graph> then gives the displayed formula.