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Counting colorful necklaces and bracelets in three colors

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Abstract

A necklace or bracelet is colorful if no pair of adjacent beads are the same color. In addition, two necklaces are equivalent if one results from the other by permuting its colors, and two bracelets are equivalent if one results from the other by either permuting its colors or reversing the order of the beads; a bracelet is thus a necklace that can be turned over. This note counts the number K(n) of non-equivalent colorful necklaces and the number \(K'(n)\) of colorful bracelets formed with n-beads in at most three colors. Expressions obtained for \(K'(n)\) simplify expressions given by OEIS sequence A114438, while the expressions given for K(n) appear to be new and are not included in OEIS.

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Authors and Affiliations

  1. Department of Aerospace Engineering, The University of Michigan, 3020 FXB Building, 1320 Beal St., Ann Arbor, MI, 48109-2140, USA

    Dennis S. Bernstein

  2. Department of Mathematics, Higher Institute for Applied Sciences and Technology, P.O. Box 31983 , Damascus, Syria

    Omran Kouba

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  1. Dennis S. Bernstein
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  2. Omran Kouba
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Correspondence to Omran Kouba.

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Bernstein, D.S., Kouba, O. Counting colorful necklaces and bracelets in three colors. Aequat. Math. 93, 1183–1202 (2019). https://doi.org/10.1007/s00010-019-00645-w

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  • DOI: https://doi.org/10.1007/s00010-019-00645-w

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