Simple formulas for the number of different cyclic and dihedral necklaces containing nj beads of the j-th color, j ≤ m and ∑m j=1 nj = N , are derived. Among a vast number of counting problems one of the most popular is a necklace enumeration. A cyclic necklace is a coloring in m colors of the vertices of a regular N–gon, where two colorings are equivalent if one can be obtained from the other ...

The Borsuk-Ulam theorem of topology is applied to a problem in discrete mathematics. A bisection of a necklace with k colors of beads is a collection of intervals whose union captures half the beads of each color. Every necklace with fc colors has a bisection formed by at most k cuts. Higherdimensional generalizations are considered.