cp's OEIS Frontend

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 exactly k parts. For n>=1, this sequence gives T(n,k) = psi_k(P_n), i.e., the number of non-equivalent distinguishing coloring partitions of the path P_n on n vertices with exactly k parts.

Also, for n > 1 the number of reversible string structures of length n using exactly k different symbols that are not equivalent to their reversal (compare A284949). - Andrew Howroyd, Aug 15 2019

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly four different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly five different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly six different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

Conjecture: 3n - a(n) is in {0, 1} for all n >= 1.

From Michel Dekking, Sep 03 2019: (Start)

Proof of the conjecture by Kimberling: more is true. Here follows a proof of the formula below. Let T be the transform T(01)=0, T(1)=0.

Consider the return word structure of A285949 for the word 1:

A285949 = 0|1000|100|10|1000|10|100| ....

[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]

The three return words are u:=10, v:=100 and w:=1000. These words uniquely correspond to the conjugated three words u'=01, v'=010, w'=0100 in A285949, which are the unique images u'=T(0), v'=T(01) and w'=T(011) of the words 0, 01 and 011 in the Thue-Morse word A010060. The images of these three words under the Thue-Morse morphism 0->01, 1->10 are 01, 0110 and 011010, and we have

T(01)=010, T(0110)=010001, T(011010)=010001001.

Shifting by 1 in A285949, these correspond uniquely to the conjugated words 100, 100010, and 100010010. It follows that the Thue-Morse morphism induces the morphism u->v, v->wu, w->wvu on the return words.

This morphism is modulo a change of alphabet equal to the ternary Thue-Morse morphism with fixed point A007413.

Note that on the alphabet {4,3,2} of the respective lengths of w, v, and u we obtain the sequence (a(n+1)-a(n)) = 4,3,2,4,2,3,4,3,2,... of first differences of the positions of the 1's in A285949.

To prove the formula a(n) = A010060(n)+ 3n-1, it suffices to show that a(n+1)-a(n) = A010060(n+1)-A010060(n)+3.

That this indeed is true: see the Comments of A029883, the first differences of the standard form of the Thue-Morse sequence A001285.

(End)

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 A(n,k) = Psi_k(P_n), i.e., the number of non-equivalent distinguishing coloring partitions of the path P_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} A309748(n,i).