Mathematical Necklaces: Symmetry, Counting, and Hidden Connections

We explore the world of mathematical necklaces: counting colorings under rotation (necklaces) and reflection (bracelets), the role of K-ary colors and length, and how these counts connect to group theory, Linden words, and irreducible polynomials. Along the way we touch on related ideas from topology and graph representations, and what happens when the counting rules are changed—revealing the surprising unity of patterns across mathematics.

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