Number of maximizing weighted k-Sperner families
= Number of maximizing weighted k-Sperner families
If $t$ is the $k$th largest rank weight $u_i$, exactly $h$ levels have weight greater than $t$, and exactly $s$ have weight $t$, then there are $\binom{s}{k-h}$ maximizing families. For fixed $n,k$, the possible counts are exactly $\binom{s}{j}$ with $1\leq j\leq k$ and $j\leq s\leq n+1-k+j$.