cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A328150 Number of n-bead necklace structures with no adjacent elements having the same color.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 12, 24, 103, 387, 1819, 8933, 48632, 279484, 1716523, 11126025, 76014437, 544945399, 4089010392, 32025053060, 261213946739, 2214280580389, 19471365925297, 177319383231697, 1669735890602062, 16235408370162588, 162796351456044465, 1681427459283678177
Offset: 0

Views

Author

Andrew Howroyd, Oct 05 2019

Keywords

Comments

Beads may be of any number of colors. Colors may be permuted without changing the necklace structure.
Equivalently, the number of set partitions of an n-set up to rotations where no block contains cyclically adjacent elements of the n-set.

Examples

			a(6) = 12 because there are the following 12 necklace structures: ABABAB, ABABAC, ABABCD, ABACAD, ABACBC, ABACBD, ABACDC, ABACDE, ABCABC, ABCABD, ABCADE, ABCDEF.

Links

Crossrefs

Row sums of A327396.
Cf. A084423.

Programs

  • PARI
    seq(n)={Vec(1 + intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(-x + sumdiv(m, d, (exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))} \\ Andrew Howroyd, Oct 09 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 09 2019

A085436 Number of partitions of n without rotational symmetry (or 1-fold symmetry).

Original entry on oeis.org

1, 0, 1, 1, 5, 2, 13, 8, 21, 17, 54, 31, 99, 70, 139, 131, 295, 207, 488, 387, 698, 657, 1253, 995, 1923, 1707, 2785, 2670, 4563, 3900, 6840, 6287, 9606, 9445, 14746, 13517, 21635, 20614, 30000, 29903, 44581, 42067, 63259
Offset: 1

Views

Author

Wouter Meeussen, Aug 14 2003

Keywords

Comments

A partition of n is considered to have d-fold symmetry if it consists of runs of (one or more) equal integers that add up to d or a multiple of n/d.
Set partitions with rotational d-fold symmetry (A084423) have block lengths that are d-fold symmetrical partitions of n, (d|n), as defined above.

Examples

			a(6)=2 since the 11 partitions of 6 consist of 4 having 6-fold symmetry: {6},{3,3},{2,2,2},{1,1,1,1,1,1}; 1 with 3-fold: {3,1,1,1}; 4 with 2-fold: {4,2},{4,1,1},{2,2,1,1},{2,1,1,1,1}; and only 2 with 1-fold symmetry (= no rotational symmetry): {5,1} and {3,2,1}.

Crossrefs

Cf. A084423.

Programs

  • Mathematica
    Needs["DiscreteMath`Combinatorica`"]; f := Function[{n, d}, Cases[ Partitions[n], q_List /; (Union[ Mod[ (First[ # ] Length[ # ] &) /@ Split[q], d]] == {0})]]; fixp[j_] := Table[d = Part[ Divisors[n], k]; Length@f[n, d], {n, j}, {k, DivisorSigma[0, n]}]; Do[ Print[ Last[ Table[ Fold[ Plus, 0, MoebiusMu[ n/ Divisors[n]] Reverse[ fixp[i][[i]] ]], {n, i}]]], {i, 1, 43}]

Extensions

Edited and extended by Robert G. Wilson v, Aug 15 2003

A211352 Refined triangle A211356: T(n,k) is the number of partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 4, 9, 3, 10, 1, 5, 3, 3, 1, 1, 3, 5, 15, 5, 30, 3, 15, 15, 10, 1, 15, 3, 5, 1, 1, 4, 7, 29, 10, 70, 7, 56, 54, 37, 4, 105, 21, 35, 1, 18, 29, 37, 4, 7, 7, 1
Offset: 1

Views

Author

Tilman Piesk, Apr 09 2012

Keywords

Comments

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,7,12,43... (A084423)
Row maxima: 1,1,1,2,3,10,30,105,420,1268,6300...
Distinct entries per row: 1,1,1,2,3,6,6,13,17,25,25...
Rightmost columns are those from the triangle of circular binomial coefficients A047996 without the second column (i.e.triangle A037306).

Links

Crossrefs

Cf. A211356, A000041, A084423, A047996, A037306.
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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