An r-ary necklace (bracelet) of length n is an equivalence class of r-colourings of vertices of a regular n-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric r-ary necklaces (bracelets) of length n is 1 2 (r + 1)r n/2 if n is even, and r (n+1)/2 if n is odd.2010 Mathematics subject classification: primary 05A15, 05C15; secondary 05C25, 11A25.
Let $n,r,k\in \mathbb{N}$. An $r$-colouring of the vertices of a regular $n$-gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$-ary necklace of length $n$ is an equivalence class of $r$-colourings of $\mathbb{Z}_{n}$. We say that a colouring is $k$-alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a $k$-alternating $r$-colouring of $\mathbb{Z}_{n}$ and we count, for any $r$, 2-alternating $r$-colourings of $\mathbb{Z}_{n}$ and 2-alternating $r$-ary necklaces of length $n$.
AbstractA semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup.
A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous.
If S is a nondiscrete monothetic semitopological semigroup, then the set {S^{\prime}} of all limit points of S is a closed ideal of S.
Let S be a locally compact nondiscrete monothetic semitopological semigroup.
We show that (1) if the translations of {S^{\prime}} are open, then {S^{\prime}} is compact, and (2) if {S^{\prime}} can be topologically and algebraically embedded in a quasitopological group, then {S^{\prime}} is a compact topological group.
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