I'm developing an algorithm that will generate a collection of subsets of a ground set having the property that no subset in the collection is a subset of any other, and I'd like to give a Big-Oh bound on the number of subsets there might be in the worst case.
From some basic research it seems that this is given by Sperner's theorem, which says that the maximum number of such subsets of an $n$-element subset is $n \choose {\lfloor{n/2}\rfloor}$. It's good to have that bound, but I don't recall ever seeing a Big-Oh expression that used the binomial coefficient, so I'm wondering if there's a reasonably tight upper bound in terms of the "usual" functions used in Big-Oh. It's clearly $\textrm O(2^n)$, but is there something tighter?
(For comparison, the "most natural" lower bound for sorting $n$ elements using comparisons is $\Omega(n!)$, but this is invariably written as $\Omega(n \log n)$, which is asymptotically equivalent on account of Stirling's approximation.)