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%I #41 Nov 05 2019 06:00:15
%S 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,
%T 1080,360,0,6,315,2484,7845,11970,8820,2520,0,14,933,10240,46280,
%U 105840,129360,80640,20160,0,30,2622,40464,254664,821592,1481760,1512000,816480,181440
%N 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.
%C The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
%C 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.
%C From _Andrew Howroyd_, Aug 15 2019: (Start)
%C First differs from A305541 at n = 6.
%C 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).
%C (End)
%H Andrew Howroyd, <a href="/A309651/b309651.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, <a href="https://arxiv.org/abs/1910.12102">Number of Distinguishing Colorings and Partitions</a>, arXiv:1910.12102 [math.CO], 2019.
%F Let n>2. For any k >= floor(n/2) we have phi_k(C_n)=k! * Stirling2(n,k)/2n.
%F T(n, k) = Sum_{i=2..k} (-1)^(k-i)*binomial(k,i)*A309528(n, i). - _Andrew Howroyd_, Aug 12 2019
%F Column k is the Moebius transform of column k of A305541. - _Andrew Howroyd_, Sep 13 2019
%e The triangle begins:
%e 0
%e 0, 0;
%e 0, 0, 1;
%e 0, 0, 3, 3;
%e 0, 0, 12, 24, 12;
%e 0, 1, 34, 124, 150, 60;
%e 0, 2, 111, 588, 1200, 1080, 360;
%e 0, 6, 315, 2484, 7845, 11970, 8820, 2520;
%e 0, 14, 933, 10240, 46280, 105840, 129360, 80640, 20160;
%e 0, 30, 2622, 40464, 254664, 821592, 1481760, 1512000, 816480, 181440;
%e ...
%e 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:
%e { { 1 }, { 2 }, { 3, 4 } }
%e { { 1 }, { 2, 3 }, { 4 } }
%e { { 1, 2 }, { 3 }, { 4 } }
%o (PARI) \\ U(n,k) is A309528
%o 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}
%o T(n,k)={sum(i=2, k, (-1)^(k-i)*binomial(k,i)*U(n,i))} \\ _Andrew Howroyd_, Aug 12 2019
%Y Columns k=2..4 are A032239(n>=3), A326660, A326789.
%Y Row sums are A326888.
%Y Cf. A001710, A254040, A309528, A305541.
%K nonn,tabl
%O 1,9
%A _Bahman Ahmadi_, Aug 11 2019
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