Let $[n]:=\lbrace 1, \dots, n \rbrace$. We define a partial ordering on the set of subsets of $[n]$ as follows. We say that $X \preceq Y$ if there is an injective map $f:X \to Y$ such that $x \leq f(x)$ for all $x \in X$. This is a pretty standard construction in poset theory.
The motivation for this question comes from a subset sum problem I've been playing with. Let us regard $[n]$ as the set of indices of a set $A:=\lbrace a_1, \dots, a_n \rbrace$ of numbers (indexed so that $a_1 < \dots < a_n$). If $X \preceq Y$, then the sum of the elements in $A$ corresponding to $X$ is at most the sum of the elements in $A$ corresponding to $Y$. If $X$ and $Y$ are incomparable, then we don't know which sum is bigger (without additional information about $A$).
I would like to cover this poset with as few chains as possible, so it is natural to apply Dilworth's Theorem and then ask
What is the size of a largest antichain in this poset?
One natural candidate is to take all subsets of $[n]$ with the same sum $s$. To maximize the size of this antichain, we should take $s$ to be halfway between $0$ and $1+\dots + n$. I'd guess that this is optimal. Any references or thoughts would be much appreciated.