Dedekind Number - Asymptotics

Asymptotics

The logarithm of the Dedekind numbers can be estimated accurately via the bounds

Here the left inequality counts the number of antichains in which each set has exactly elements, and the right inequality was proven by Kleitman & Markowsky (1975).

Korshunov (1981) provided the even more accurate estimates

for even n, and

for odd n, where

and

The main idea behind these estimates is that, in most antichains, all the sets have sizes that are very close to n/2. For n = 2, 4, 6, 8 Korshunov's formula provides an estimate that is inaccurate by a factor of 9.8%, 10.2%, 4.1%, and -3.3%, respectively.