Calling this poset $M(n)$, the fact that it has the <a href="http://en.wikipedia.org/wiki/Sperner_property_of_a_partially_ordered_set">Sperner property</a> was conjectured in B. Lindström, "A conjecture on a theorem similar to Sperner's", Combinatorial Structures and Their Applications, p. 241.

It turns out that $M(n)$ has the $k$-Sperner property for all $k$, see R. Stanley's paper "Weyl groups, the hard Lefschetz theorem, and the Sperner property", in the section on the type $B_n$ the properties of $M(n)$ are shown to follow from the general theorems about posets derived from complex semisimple algebraic groups by quotienting by a parabolic subgroup and using the Bruhat order. See also Stanley's article "Some applications of algebra to combinatorics". The papers are available at <a href="http://www-math.mit.edu/~rstan/pubs/">his website</a>.