Asymptotic Enumeration of Partial Orders on a Finite Set
D. J. Kleitman, Bruce L. Rothschild · Transactions of the American Mathematical Society · 1975
By considering special cases, the number ${P_n}$ of partially ordered sets on a set of $n$ elements is shown to be $(1 + O(1/n)){Q_n}$, where ${Q_n}$ is the number of partially ordered sets in one of the special classes. The number ${Q_n}$ can be estimated, and we ultimately obtain \[ {P_n} = \left ( {1 + O\left ( {\frac {1}{n}} \right )} \right )\left ( {\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{n - i} {\left ( {_i^n} \right )\left ( {_j^{n - i}} \right ){{\left ( {{2^i} - 1} \right )}^j}{{\left ( {{2^j} - 1} \right )}^{n - i - j}}} } } \right ).\]