On Dedekind’s problem: The number of monotone Boolean functions
Daniel J. Kleitman · Proceedings of the American Mathematical Society · 1969
The problem of determining the number \p(n) of elements of the free distributive lattice on « generators was posed by Dedekind [l ] in 1897.It was solved by that author for « = 4. R. Church [2] in 1940 and M. Ward [3] in 1946 obtained solutions for « = 5 and w = 6 respectively.In 1954 E. N. Gilbert [4] showed that^(w) satisfied the inequalities 'fn.lnin = ^,(w) <; wCn,[n/2]+2 (where Cn,[nß) is the binomial coefficient).Korobkov [5] in several papers published in 1962-1965 was able to improve the upper bound in \p(n) to 4.23C 2 ».[n/2]iIn 1966 G. Hansel [ó] reduced the upper bound still further to 3Cn.[n/2]_In this paper we show that log2^(«) is asymptotic to Cn,inn\', m fact we show that 2<l+a»)Cn, [n/2] <; ^,(M) _ 2 U+ßn)Cn, [n/2]