The answer to the updated question is to take the sum of the $k-1$ middle bịnomial coefficients. This was first proved by Erdos in this paper (see Theorem 5).

To address the updated question, a family of subsets of $[n]$ is called $k$-Sperner if it does not contain a chain of length $k+1$. By taking all sets whose size lies in the middle $k$ values of $[n]$, there exist $k$-Sperner families who size is the sum of the $k$ middle bịnomial coefficients. Erdős proved that this bound is tight in this paper (see Theorem 5). The extremal example is also essentially unique (for parity reasons there may be two intervals of middle $k$ values).

The answer to the updated question is to take the sum of the k-1 middle bịnomial coefficients. This was first proved by Erdos in this (paper]) [https://projecteuclid.org/euclid.bams/1183507531] (see Theorem 5).

The answer to the updated question is to take the sum of the $k-1$ middle bịnomial coefficients. This was first proved by Erdos in this paper (see Theorem 5).