Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2003-19

November 2003


The Number of Linear Extensions of the Boolean Lattice

Graham Brightwell and Prasad Tetali

Abstract

Let 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

\begin{displaymath}
\frac{\log(L(Q^t))}{2^t} =
\log{t \choose \lfloor{t/2}\rfloor} - \frac{3}{2}\log e + o(1),
\end{displaymath}

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.


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