On Dedekind’s problem: the number of isotone Boolean functions. II
Open Access
- 1 January 1975
- journal article
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 213, 373-390
- https://doi.org/10.1090/s0002-9947-1975-0382107-0
Abstract
It is shown that ψ ( n ) \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 \[ ψ ( n ) ⩽ 2 ( 1 + O ( log n / n ) ) ( n [ n / 2 ] ) . \psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left ( {\begin {array}{*{20}{c}} n \\ {[n/2]} \\ \end {array} } \right )}}. \] This result is an improvement by a factor n \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.Keywords
This publication has 2 references indexed in Scilit:
- The asymptotic number of geometriesJournal of Combinatorial Theory, Series A, 1974
- On Dedekind's Problem: The Number of Monotone Boolean FunctionsProceedings of the American Mathematical Society, 1969