Edge boundary of a down-set in a cube ($b_e(D)=n|D|-2{\sum }_{A\in D}|A|$)

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.

For a down-set of size $m$, maximize its edge boundary by minimizing the sum of the sizes of its members. Choose all sets in the smallest ranks, followed by any needed part of the next rank; this minimizes that sum among all families and is itself a down-set. With $M_j={\sum }_{i=0}^j\left(\genfrac{}{}{0ex}{}{n}{i}\right)$, $M_{-1}=0$, and $M_{r-1}\le m\le M_r$, the exact maximum is $h(m)=nm-2[{\sum }_{j<r}j(\genfrac{}{}{0ex}{}{n}{j})+r(m-M_{r-1})]$.