The Number of Linear Extensions of the Boolean Lattice

AbstractLet L(Qt) denote the number of linear extensions of the t-dimensional Boolean lattice Qt. We use the entropy method of Kahn to show that $$\frac{{\log (L(Q^t ))}}{{2^t }} = \log \left(\begin{gathered} t \hfill \\ \left| \!{\underline {\, t \,}} \right. /\left. {\underline {\, 2 \,}}\! \right| \hfill \\ \end{gathered} \right) - \frac{3} {2}\log e + o(1),$$ where the logarithms are base 2. We also find the exact maximum number of linear extensions of a d-regular bipartite order on n elements, in the case when n is a multiple of 2d.