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A309748
The number of non-equivalent distinguishing coloring partitions of the path on n vertices (n>=1) with exactly k parts (k>=1). Regular triangle read by rows: the rows are indexed by n, the number of vertices of the path, and the columns are indexed by k, the number of parts.
7
1, 0, 1, 0, 1, 1, 0, 4, 4, 1, 0, 6, 14, 6, 1, 0, 16, 49, 37, 9, 1, 0, 28, 154, 182, 76, 12, 1, 0, 64, 496, 876, 542, 142, 16, 1, 0, 120, 1520, 3920, 3522, 1346, 242, 20, 1, 0, 256, 4705, 17175, 21392, 11511, 2980, 390, 25, 1, 0, 496, 14266, 73030, 123665, 89973, 32141, 5990, 595, 30, 1
OFFSET
1,8
COMMENTS
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
LINKS
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
Mohammad Hadi Shekarriz, GAP Program
FORMULA
T(n,k) = A309635(n,k) - A309635(n,k-1) for k > 1.
T(n,k) = A284949(n,k) - Stirling2(ceiling(n/2), k) for n > 1. - Andrew Howroyd, Aug 15 2019
EXAMPLE
The triangle begins:
1;
0, 1;
0, 1, 1;
0, 4, 4, 1;
0, 6, 14, 6, 1;
0, 16, 49, 37, 9, 1;
0, 28, 154, 182, 76, 12, 1;
0, 64, 496, 876, 542, 142, 16, 1;
0, 120, 1520, 3920, 3522, 1346, 242, 20, 1;
0, 256, 4705, 17175, 21392, 11511, 2980, 390, 25, 1;
...
----
For n=4, we can partition the vertices of P_4 into exactly 3 parts in 4 ways such that all these partitions induce distinguishing colorings for P_4 and that all the 4 partitions are non-equivalent. The partitions are as follows:
{ { 1 }, { 2 }, { 3, 4 } }
{ { 1 }, { 2, 3 }, { 4 } }
{ { 1 }, { 2, 4 }, { 3 } }
{ { 1, 4 }, { 2 }, { 3 } }
PROG
(PARI) \\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
T(n)={(matrix(n, n, i, k, stirling(i, k, 2) - 2*stirling((i+1)\2, k, 2)) + Ach(n))/2}
{ my(A=T(10)); A[1, 1]=1; for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 18 2019
CROSSREFS
Columns k=2..4 are A007179, A327610, A327611.
Row sums are A327612(n > 1).
Sequence in context: A373428 A163353 A164612 * A180401 A354043 A057270
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Terms a(56) and beyond from Andrew Howroyd, Sep 18 2019
STATUS
approved