Regarding the number of partial orders $P_n$ on an $n$-set

Question

Regarding the set of partial orders $P_n$ on an $n$-set, it is known that $$|P_n|\geq {2^\frac{n^2}{4}}$$ If I replace $n$ by $2n$, I get $$|P_{2n}|\geq 2^{n^2}$$ So, if a $2n$-set, say $A=\{1,2,3,\cdots,2n\}$ is partitioned into two sets $A_1=\{1,2,3,\cdots,n\}$ and $A_2=\{n+1,n+2,n+3,\cdots,2n\}$, then doesn't it mean there there exists a function $$f:\mathcal{P}(A_1\times A_1)\to P_{2n}$$ such that $f$ is one-one, $\mathcal{P(A)}$ being the symbol for power set of $\mathcal{A}$? If yes, how would that function $f$ be constructed?