Asymptotic Enumeration of Partial Orders on a Finite Set

By considering special cases, the number of partially ordered sets on a set of elements is shown to be <!-- MATH $(1 + O(1/n)){Q_n}$ --> , where is the number of partially ordered sets in one of the special classes. The number can be estimated, and we ultimately obtain <!-- MATH \begin{displaymath} {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). \end{displaymath} -->