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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 (list; table; graph; refs; listen; history; text; internal format)
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

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

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).

Cf. A284949, A304972, A309635, A309784.

Sequence in context: A336302 A163353 A164612 * A180401 A057270 A057278

Adjacent sequences:  A309745 A309746 A309747 * A309749 A309750 A309751

KEYWORD

nonn,tabl

AUTHOR

Mohammad Hadi Shekarriz, Aug 15 2019

EXTENSIONS

Terms a(56) and beyond from Andrew Howroyd, Sep 18 2019

STATUS

approved

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Last modified January 26 07:38 EST 2022. Contains 350577 sequences. (Running on oeis4.)