OEIS A000370: NPN Equivalence Classes of Boolean Functions

NPN equivalence groups functions that can be turned into one another by flipping inputs, permuting inputs, and possibly inverting the output. A000370 counts how many such equivalence classes remain for each n: 1 for n=0, 2 for n=1, 4 for n=2, 14 for n=3, 222 for n=4, and 616,126 for n=5, illustrating the dramatic compression. In practice, each class has a canonical representative—the lexicographically smallest truth table among all NPN transforms—so tools can store one circuit per class and realize others by simple wiring or inverters. We’ll unpack the group action that does the flipping and swapping, why the reduction is so powerful, and what Harrison proved about the asymptotic growth of the number of classes. It’s a striking example of how structure hides in plain sight in Boolean logic and why researchers study these symmetries—it's a shortcut through combinatorial chaos.

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