A327693 -id:A327693 - cp's OEIS frontend

A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 1, 12, 17, 4, 0, 0, 0, 2, 43, 82, 49, 9, 0, 0, 0, 7, 137, 388, 339, 125, 15, 0, 0, 0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0, 0, 31, 1190, 6405, 10900, 7928, 2761, 490, 35, 0, 0, 0, 57, 3394, 24853, 56586, 54272, 25609, 6365, 850, 51, 0, 0

Offset: 1

Bahman Ahmadi, Sep 04 2019

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with exactly k parts.
Number of n-bead bracelet structures using exactly k different colored beads that are not self-equivalent under either a nonzero rotation or reversal (turning over bracelet). Comparable sequences for unoriented (reversible) strings and necklaces (cyclic group) are A320525 and A327693. - Andrew Howroyd, Sep 23 2019

			Triangle begins:
  0;
  0,  0;
  0,  0,   0;
  0,  0,   0,    0;
  0,  0,   0,    0,    0;
  0,  0,   4,    2,    0,    0;
  0,  1,  12,   17,    4,    0,   0;
  0,  2,  43,   82,   49,    9,   0,  0;
  0,  7, 137,  388,  339,  125,  15,  0, 0;
  0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0;
  ...
For n=7, we can partition the vertices of the cycle C_7 with exactly 3 parts, in 12 ways, such that all these partitions are distinguishing for C_7 and that all the 12 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2, 3 }, { 4, 5, 6, 7 } },
    { { 1 }, { 2, 3, 4, 6 }, { 5, 7 } },
    { { 1 }, { 2, 3, 4, 7 }, { 5, 6 } },
    { { 1 }, { 2, 3, 5, 6 }, { 4, 7 } },
    { { 1 }, { 2, 3, 5, 7 }, { 4, 6 } },
    { { 1 }, { 2, 3, 6 }, { 4, 5, 7 } },
    { { 1 }, { 2, 3, 7 }, { 4, 5, 6 } },
    { { 1 }, { 2, 4, 5, 6 }, { 3, 7 } },
    { { 1 }, { 2, 4, 7 }, { 3, 5, 6 } },
    { { 1, 2 }, { 3, 4, 6 }, { 5, 7 } },
    { { 1, 2 }, { 3, 5, 6 }, { 4, 7 } },
    { { 1, 2, 4 }, { 3, 6 }, { 5, 7 } }.
From _Andrew Howroyd_, Sep 23 2019: (Start)
For n=6, k=4 the partitions are:
    { { 1, 2, 4 }, { 3 }, { 5 }, { 6 } },
    { { 1, 2 }, { 3, 5 }, { 4 }, { 6 } }.
These correspond to the bracelet structures AABACD and AABCBD.
(End)

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

Column k=2 is A327734.
Row sums are A327740.
Cf. A309784, A309785, A320525, A320748, A152176, A320647, A324803, A327693.

T(n,k) = A324803(n,k) - A324803(n,k-1).

a(56)-a(78) from Andrew Howroyd, Sep 23 2019

A328740 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly three different colored beads.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 43, 116, 335, 920, 2591, 7173, 20125, 56260, 158333, 446098, 1262225, 3579227, 10181479, 29028405, 82968695, 237634700, 682014587, 1960951980, 5647919640, 16292680600, 47069103545, 136166476875, 394418199725, 1143821408316, 3320790074371

Offset: 1

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

			For n=5, the 5 necklace structures are: aaabc, aabac, aabbc, aabcb, ababc.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 3 of A327693.

Cf. A056296, A056304, A328741, A328742.

a(p) = A056296(p) = A056304(p) for prime p >= 5.

A328741 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 2, 9, 50, 206, 862, 3384, 13250, 50852, 194810, 741884, 2823712, 10735998, 40843370, 155483873, 592614050, 2261585918, 8643288808, 33080772468, 126797503250, 486710415710, 1870851589552, 7201012694120, 27752927349880, 107092389688830, 413729680838330

Offset: 1

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

			For n=6, the 9 necklace structures are: aaabcd, aabacd, aabcad, aabbcd, aabcbd, aabcdb, aacbdb, ababcd, abacbd.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 4 of A327693.

Cf. A056297, A056305, A328740, A328743.

a(p) = A056297(p) = A056305(p) for prime p.

A327696 Number of n-bead necklace structures using an infinite alphabet that are not self-equivalent under a nonzero rotation.

Original entry on oeis.org

1, 1, 0, 1, 2, 10, 28, 125, 497, 2345, 11497, 61688, 350543, 2126495, 13631966, 92197099, 654982126, 4874404106, 37892942311, 306986431845, 2586206074833, 22612848380851, 204850698287662, 1919652428481928, 18581619383614327, 185543613289199731, 1908894095258021121

Offset: 0

Andrew Howroyd, Sep 22 2019

nonn

Permuting the colors does not change the structure.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A327693.

Cf. A084423, A276547.

seq(n)={Vec(1 + intformal(sum(m=1, n, moebius(m)*subst(serlaplace(-1 + exp(sumdiv(m, d, (exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))}

A328742 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using a maximum of three different colored beads.

Original entry on oeis.org

1, 1, 0, 1, 2, 8, 17, 52, 130, 363, 968, 2684, 7338, 20440, 56836, 159424, 448130, 1266080, 3586479, 10195276, 29054568, 83018624, 237729932, 682196948, 1961301330, 5648590728, 16293970840, 47071589049, 136171269780, 394427456120, 1143839302904, 3320824711204

Offset: 0

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A002075, A002076, A327693, A328740.

a(n) = Sum_{k=1..3} A327693(n, k) for n > 0.

a(p) = A002075(p) for prime p >= 5.

A328743 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using a maximum of four different colored beads.

Original entry on oeis.org

1, 1, 0, 1, 2, 10, 26, 102, 336, 1225, 4352, 15934, 58190, 215250, 798720, 2983136, 11184128, 42109450, 159070352, 602809326, 2290640486, 8726307432, 33318502400, 127479700198, 488671717040, 1876500180280, 7217306664960, 27799998938929, 107228560958610, 414124108294450

Offset: 0

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A056292, A056300, A327693, A328741.

a(n) = Sum_{k=1..4} A327693(n, k) for n > 0.

a(p) = A056300(p) for prime p >= 5.

cp's OEIS Frontend

A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

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A328740 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly three different colored beads.

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A328741 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly four different colored beads.

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A327696 Number of n-bead necklace structures using an infinite alphabet that are not self-equivalent under a nonzero rotation.

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A328742 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using a maximum of three different colored beads.

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A328743 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using a maximum of four different colored beads.

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