A328130 -id:A328130 - cp's OEIS frontend

A327396 Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.

0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 3, 5, 2, 1, 0, 0, 3, 10, 8, 2, 1, 0, 1, 7, 33, 40, 18, 3, 1, 0, 0, 11, 83, 157, 104, 28, 3, 1, 0, 1, 19, 237, 650, 615, 246, 46, 4, 1, 0, 0, 31, 640, 2522, 3318, 1857, 495, 65, 4, 1, 0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1

Offset: 1

Andrew Howroyd, Oct 04 2019

Permuting the colors does not change the necklace structure.

Equivalently, the number of k-block partitions of an n-set up to rotations where no block contains cyclically adjacent elements of the n-set.

			Triangle begins:
  0;
  0, 1;
  0, 0,  1;
  0, 1,  1,    1;
  0, 0,  1,    1,    1;
  0, 1,  3,    5,    2,     1;
  0, 0,  3,   10,    8,     2,     1;
  0, 1,  7,   33,   40,    18,     3,    1;
  0, 0, 11,   83,  157,   104,    28,    3,   1;
  0, 1, 19,  237,  650,   615,   246,   46,   4,  1;
  0, 0, 31,  640, 2522,  3318,  1857,  495,  65,  4, 1;
  0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1;
  ...

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

Columns k=3..4 are A327397, A328130.
Partial row sums include A306888, A309673.
Row sums are A328150.
Cf. A152175, A261139, A208535.

R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace((y-1)*exp(-x + O(x*x^(n\m))) - y + exp(-x + sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d)) ), x, x^m))/x), -n)]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 09 2019

A309673 Number of n-bead necklace structures using a maximum of four different colored beads and no adjacent beads having the same color.

Original entry on oeis.org

0, 1, 1, 3, 2, 9, 13, 41, 94, 257, 671, 1881, 5110, 14301, 39871, 112281, 316520, 897297, 2548819, 7265383, 20754748, 59437181, 170549237, 490338539, 1412147684, 4073528481, 11767897903, 34042917197, 98606864030, 285960106473, 830206177801, 2412787265021

Offset: 1

Andrew Howroyd, Oct 05 2019

nonn

Colors may be permuted without changing the necklace structure.

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

Cf. A056292, A306888, A327396, A328130.

a(n) = Sum_{k=1..4} A327396(n, k).

Terms a(24) and beyond from Andrew Howroyd, Oct 10 2019

A327397 Number of n-bead necklace structures with beads of exactly three colors and no adjacent beads having the same color.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 3, 7, 11, 19, 31, 63, 105, 201, 367, 695, 1285, 2451, 4599, 8775, 16651, 31837, 60787, 116639, 223697, 430395, 828525, 1598227, 3085465, 5965999, 11545611, 22370999, 43383571, 84217615, 163617805, 318150719, 619094385, 1205614053, 2349384031

Offset: 1

Andrew Howroyd, Oct 04 2019

nonn

Colors may be permuted without changing the necklace structure.

			Necklace structures for n=3..8 are:
  a(3) = 1: ABC;
  a(4) = 1: ABAC;
  a(5) = 1: ABABC;
  a(6) = 3: ABABAC, ABACBC, ABCABC;
  a(7) = 3: ABABABC, ABABCAC, ABACABC;
  a(8) = 7: ABABABAC, ABABACAC, ABABACBC, ABABCABC, ABABCBAC, ABACABAC, ABACBABC.

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

Column k=3 of A327396.

Cf. A056296, A306888, A328130.

a(n) = A306888(n) - A000035(n-1). - Yuchun Ji, Mar 13 2020

Terms a(33) and beyond from Andrew Howroyd, Oct 09 2019

cp's OEIS Frontend

A327396 Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.

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A309673 Number of n-bead necklace structures using a maximum of four different colored beads and no adjacent beads having the same color.

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A327397 Number of n-bead necklace structures with beads of exactly three colors and no adjacent beads having the same color.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions