On Dedekind’s problem: the number of isotone Boolean functions. II

Kleitman, D.; Markowsky, G.

doi:10.1090/S0002-9947-1975-0382107-0

On Dedekind’s problem: the number of isotone Boolean functions. II
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by D. Kleitman and G. Markowsky PDF

Trans. Amer. Math. Soc. 213 (1975), 373-390 Request permission

Abstract:

It is shown that $\psi (n)$, the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies \[ \psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left ( {\begin {array}{*{20}{c}} n \\ {[n/2]} \\ \end {array} } \right )}}.\] This result is an improvement by a factor $\sqrt n$ in the 0 term of a previous result of Kleitman. In the course of deriving the main result, we analyze thoroughly the techniques used here and earlier by Kleitman, and show that the result in this paper is “best possible” (up to constant) using these techniques.

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Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices