OEIS A000133: Boolean Functions, Burnside, and the Geometry of Symmetry

A deep dive into A000133, the OEIS entry counting Boolean functions of n variables up to natural symmetries. We start from truth tables and show how Burnside's lemma trims the combinatorial explosion from 2^(n·2^n) possibilities into a small, meaningful count (e.g., five for n=2, thirty for n=3, with numbers soaring for larger n). Along the way we connect these counts to elementary abelian groups and their subgroups, and to Boolean rings and ideals—illuminating how a topic in logic ties into core ideas in algebra and number theory. We’ll also touch on the historical lineage of Burnside's lemma and why these ideas resonate with number theory students as a concrete bridge between combinatorics, group actions, and algebra.

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