A309651 T(n,k) is the number of non-equivalent distinguishing colorings of the cycle on n vertices with exactly k colors (k>=1). Regular triangle read by rows, n >= 1, 1 <= k <= n.

0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 34, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 315, 2484, 7845, 11970, 8820, 2520, 0, 14, 933, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2622, 40464, 254664, 821592, 1481760, 1512000, 816480, 181440

Offset

1,9

Comments

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).

(End)

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Formula

Let n>2. For any k >= floor(n/2) we have phi_k(C_n)=k! * Stirling2(n,k)/2n.

T(n, k) = Sum_{i=2..k} (-1)^(k-i)*binomial(k,i)*A309528(n, i). - Andrew Howroyd, Aug 12 2019

Column k is the Moebius transform of column k of A305541. - Andrew Howroyd, Sep 13 2019

Example

The triangle begins:
  0
  0,  0;
  0,  0,    1;
  0,  0,    3,     3;
  0,  0,   12,    24,     12;
  0,  1,   34,   124,    150,     60;
  0,  2,  111,   588,   1200,   1080,     360;
  0,  6,  315,  2484,   7845,  11970,    8820,    2520;
  0, 14,  933, 10240,  46280, 105840,  129360,   80640,  20160;
  0, 30, 2622, 40464, 254664, 821592, 1481760, 1512000, 816480, 181440;
  ...
For n=4, we can color the vertices of the cycle C_4 with exactly 3 colors, in 3 ways, such that all the colorings distinguish the graph (i.e., no non-identity automorphism of C_4 preserves the coloring) and that all the three colorings are non-equivalent. The color classes are as follows:
{ { 1 }, { 2 }, { 3, 4 } }
{ { 1 }, { 2, 3 }, { 4 } }
{ { 1, 2 }, { 3 }, { 4 } }

Prog

(PARI) \\ U(n, k) is A309528
U(n, k)={sumdiv(n, d, moebius(n/d)*(k^d/n - if(d%2, k^((d+1)/2), (k+1)*k^(d/2)/2)))/2}
T(n, k)={sum(i=2, k, (-1)^(k-i)*binomial(k, i)*U(n, i))} \\ Andrew Howroyd, Aug 12 2019

Crossrefs

Columns k=2..4 are A032239(n>=3), A326660, A326789.

Row sums are A326888.

Cf. A001710, A254040, A309528, A305541.

Sequence in context: A060523 A199041 A199237 * A305541 A280810 A283386

Adjacent sequences: A309648 A309649 A309650 * A309652 A309653 A309654

Keyword

nonn,tabl

Author

Bahman Ahmadi, Aug 11 2019

Status

approved