$p(n)\,=\,2^{n^2/4+3n/2+O(\log_2n)}$
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An antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable.
They are the elements from the first set in the original Carestian product. For example, if you make ordered pairs on an x-y plane, then they are the elements of the set X.
It is called an ordered pair.
it denotes the set of ordered pairs with elements of A and b in the format (a,b)
If the set has n elements, the number of subsets (the power set) has 2n members.