Counting Antichains and Linear Extensions in Generalizations of the Boolean Lattice

In 1897, Dedekind [6] posed the problem of estimating the number of antichains in the Boolean lattice; in particular, he asked whether the logarithm of the number is asymptotic to the size of the middle layer of the n-dimensional Boolean lattice Bn. Although Kleitman confirmed the truth of this conjecture in 1969 [13], enumerating antichains in Bn has continued to generate interest in the mathematical and computer science communities ([14], [17], [11], ...), culminating in the works of Korshunov [15] and Sapozhenko [19] who found sharp estimates on the actual number of antichains (rather than producing results at the logarithmic level). Note the order of these results: although asymptotics for the number of antichains was known in 1980 subsequent research provided estimates which were less accurate. Although language barriers may have contributed to this progression of results, (some of the seminal papers have not been translated from their original Russian), there are other relevant factors. The proofs of these sharp estimates are very complicated and involve intense case analysis. Counting antichains is a problem well suited to modern entropy-based enumeration techniques, since information about local properties, such as vertex degrees, can be translated into a global property of the poset. Entropy based enumeration proofs are typically beautiful and succinct, with the tradeoff that the results are for the logarithm of the number of objects one is trying to count. A similarly compelling question was raised by Stanley (see [20]) and others independently: How many linear extensions of the Boolean Lattice can be formed?