J. Gorodkin

Discrete Applied Mathematics 75, 269-275, 1997.

** Abstract **

The group of congruences and permutations of the two-colored
*N*-dimensional Boolean cube is considered. The total number of orbits
generated by these automorphisms are shown to scale as
2^{2N}/(2^{N+1}*N*!) when
*N* tends to infinity. The probability that a randomly chosen function
will belong to an orbit containing the maximum possible number of elements,
2^{N+1}*N*!, approaches one as *N* goes to infinity.
Simulations for *N*<=6 are in agreement with the scaling predictions.

*Keyword(s)*: Boolean functions; Orbits; Equivalence classes; Counting;
Neural networks