Counting inequivalent monotone Boolean functions

Monotone Boolean functions (MBFs) are Boolean functions f:{0,1}n→{0,1}f:{0,1}n→{0,1} satisfying the monotonicity condition x≤y⇒f(x)≤f(y)x≤y⇒f(x)≤f(y) for any x,y∈{0,1}nx,y∈{0,1}n. The number of MBFs in nn variables is known as the nnth Dedekind number. It is a longstanding computational challenge to determine these numbers exactly: these values are only known for nn at most 8. Two monotone Boolean functions are equivalent if one can be obtained from the other by permuting the variables. The number of inequivalent MBFs in nn variables was known only for up to n=6n=6. In this paper we propose a strategy to count inequivalent MBFs by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148.