cp's OEIS Frontend

The cycle graph is defined for n >= 3; extended to n=1,2 using the closed form.

Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=Xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with at most k parts.

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.

Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with exactly k parts.

Number of n-bead bracelet structures using exactly k different colored beads that are not self-equivalent under either a nonzero rotation or reversal (turning over bracelet). Comparable sequences for unoriented (reversible) strings and necklaces (cyclic group) are A320525 and A327693. - Andrew Howroyd, Sep 23 2019

A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called equivalent if there is an automorphism of the graph which preserves the colors of the vertices. Given a graph G, we use the notation Phi_k(G) to denote the number of non-equivalent distinguishing colorings of G with at most k colors. The sequence here, displays A(n,k)=Phi_k(C_n), i.e., the number of non-equivalent distinguishing colorings of the cycle C_n on n vertices with at most k colors.

A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. A distinguishing coloring partition of a graph G is a partition of the vertices of G such that it induces a distinguishing coloring for G. We say two distinguishing coloring partitions P1 and P2 of G are equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. Given a graph G, we use the notation Psi_k(G) to denote the number of non-equivalent distinguishing coloring partitions of G with at most k parts. This sequence gives T(n,k) = Psi_k(C_n), i.e., the number of non-equivalent distinguishing coloring partitions of the cycle C_n on n vertices with at most k parts.

Note that, for any graph G, Psi_k(G) = Sum_{i<=k} psi_i(G), where psi_i(G) is the number of non-equivalent distinguishing coloring partitions of G with exactly i parts. For instance, here we have T(n,k) = Sum_{i<=k} A309784(n,i).

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.

A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called equivalent if there is an automorphism of the graph which preserves the colors of the vertices. Given a graph G, we use the notation phi_k(G) to denote the number of non-equivalent distinguishing colorings of G with exactly k colors. The sequence here, displays T(n,k)=phi_k(C_n), i.e., the number of non-equivalent distinguishing colorings of the cycle C_n on n vertices with exactly k colors.

From Andrew Howroyd, Aug 15 2019: (Start)

First differs from A305541 at n = 6.

Also the number of n-bead asymmetric bracelets with exactly k different colored beads. More precisely the number of chiral pairs of primitive (aperiodic) color loops of length n with exactly k different colors. For example, for n=4 and k = 3, there are 3 achiral loops (1213, 1232, 1323) and 3 pairs of chiral loops (1123/1132, 1223/1322, 1233/1332).

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