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%I A309785 #25 Nov 05 2019 06:00:25
%S A309785 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,2,4,1,0,0,0,1,2,6,9,
%T A309785 1,0,0,0,1,2,7,19,26,4,0,0,0,1,2,7,22,58,66,7,0,0,0,1,2,7,23,74,195,
%U A309785 183,18,0,0,0,1,2,7,23,77,279,651,488,31,0
%N A309785 The number of non-equivalent distinguishing coloring partitions of the cycle on n vertices with at most k parts (k>=1). The cycle graph is defined for n>=3; extended to n=1,2 using the closed form. Square array read by descending antidiagonals: the rows are indexed by n, the number of vertices of the cycle, and the columns are indexed by k, the number of parts.
%C A309785 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.
%C A309785 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).
%H A309785 Bahman Ahmadi, <a href="/A309785/a309785.txt">GAP Program</a>
%H A309785 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 A309785 T(n,k) = Sum_{i=1..k} A309784(n,i).
%e A309785 Table begins:
%e A309785 =============================================================
%e A309785 n\k| 1 2 3 4 5 6 7 8 9 10
%e A309785 ---+---------------------------------------------------------
%e A309785 1 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ...
%e A309785 2 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ...
%e A309785 3 | 0, 0, 1, 1, 1, 1, 1, 1, 1, 1 ...
%e A309785 4 | 0, 0, 1, 2, 2, 2, 2, 2, 2, 2 ...
%e A309785 5 | 0, 0, 4, 6, 7, 7, 7, 7, 7, 7 ...
%e A309785 6 | 0, 1, 9, 19, 22, 23, 23, 23, 23, 23 ...
%e A309785 7 | 0, 1, 26, 58, 74, 77, 78, 78, 78, 78 ...
%e A309785 8 | 0, 4, 66, 195, 279, 306, 310, 311, 311, 311 ...
%e A309785 9 | 0, 7, 183, 651, 1084, 1255, 1292, 1296, 1297, 1297 ...
%e A309785 10 | 0, 18, 488, 2294, 4554, 5802, 6140, 6194, 6199, 6200 ...
%e A309785 ...
%e A309785 -------
%e A309785 For n=6, we can partition the vertices of C_6 into at most 4 parts in 10 ways such that all these partitions induce distinguishing colorings for C_6 and that all the 10 partitions are non-equivalent.
%e A309785 { { 1 }, { 2 }, { 3 }, { 4, 5, 6 } }
%e A309785 { { 1 }, { 2 }, { 3, 4 }, { 5, 6 } }
%e A309785 { { 1 }, { 2 }, { 3, 4, 6 }, { 5 } }
%e A309785 { { 1 }, { 2 }, { 3, 5 }, { 4, 6 } }
%e A309785 { { 1 }, { 2 }, { 3, 6 }, { 4, 5 } }
%e A309785 { { 1 }, { 2, 3 }, { 4 }, { 5, 6 } }
%e A309785 { { 1 }, { 2, 3 }, { 4, 6 }, { 5 } }
%e A309785 { { 1 }, { 2, 4 }, { 3, 6 }, { 5 } }
%e A309785 { { 1 }, { 2, 4, 6 }, { 3 }, { 5 } }
%e A309785 { { 1 }, { 2, 5 }, { 3, 6 }, { 4 } }
%Y A309785 Cf. A309635, A309784.
%K A309785 nonn,tabl
%O A309785 1,25
%A A309785 _Bahman Ahmadi_, Aug 17 2019