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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A323862 Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.

Original entry on oeis.org

2, 2, 2, 6, 10, 6, 12, 54, 54, 12, 30, 228, 498, 228, 30, 54, 990, 4020, 4020, 990, 54, 126, 3966, 32730, 65040, 32730, 3966, 126, 240, 16254, 261522, 1047540, 1047540, 261522, 16254, 240, 504, 65040, 2097018, 16768860, 33554370, 16768860, 2097018, 65040, 504
Offset: 1

Views

Author

Gus Wiseman, Feb 04 2019

Keywords

Comments

A sequence of length n is aperiodic if all n rotations of its entries are distinct.

Examples

			Array begins:
        2        2        6       12       30
        2       10       54      228      990
        6       54      498     4020    32730
       12      228     4020    65040  1047540
       30      990    32730  1047540 33554370

Links

Crossrefs

First and last columns are A027375. Main diagonal is A265627.
Cf. A000031, A000740, A001037, A179043, A323351.
Cf. A323860, A323861, A323863, A323864, A323865, A323867, A323869.

Programs

  • Mathematica
    nn=5;
a[n_,k_]:=Sum[MoebiusMu[d]*MoebiusMu[e]*2^(n/d*k/e),{d,Divisors[n]},{e,Divisors[k]}];
Table[a[n-k,k],{n,nn},{k,n-1}]
  • PARI
    A(n,k) = {sumdiv(n, d, sumdiv(k,e, moebius(d) * moebius(e) * 2^((n/d) * (k/e))))} \\ Andrew Howroyd, Jan 19 2023

Formula

A(n,k) = Sum_{d|n, e|k} mu(d) * mu(e) * 2^((n/d) * (k/e)).

A323872 Number of n X n aperiodic binary toroidal necklaces.

Original entry on oeis.org

1, 2, 2, 54, 4050, 1342170, 1908852102, 11488774559598, 288230375950387200, 29850020237398244599296, 12676506002282260237970435130, 21970710674130840874443091905460038, 154866286100907105149455216472736043777350, 4427744605404865645682169434028029029963535277450
Offset: 0

Views

Author

Gus Wiseman, Feb 04 2019

Keywords

Comments

The 1-dimensional (Lyndon word) case is A001037.
We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

Examples

			Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
  [0 0] [0 1]
  [0 1] [1 1]
Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
  000  000  000  000  000  000  000  000  000
  000  000  001  001  001  001  001  001  001
  001  011  001  010  011  100  101  110  111
.
  000  000  000  000  000  000  000  000  000
  011  011  011  011  011  011  011  111  111
  001  010  011  100  101  110  111  001  011
.
  001  001  001  001  001  001  001  001  001
  001  001  001  001  001  001  010  010  010
  010  011  100  101  110  111  011  101  110
.
  001  001  001  001  001  001  001  001  001
  010  011  011  011  011  011  100  100  100
  111  010  011  101  110  111  011  110  111
.
  001  001  001  001  001  001  001  001  001
  101  101  101  101  110  110  110  110  111
  011  101  110  111  011  101  110  111  011
.
  001  001  001  011  011  011  011  011  011
  111  111  111  011  011  011  101  110  111
  101  110  111  101  110  111  111  111  111

Links

Crossrefs

Main diagonal of A323861.
Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.
Cf. A323859, A323860, A323865, A323866, A323871.

Programs

  • Mathematica
    apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]

Extensions

Terms a(5) and beyond from Andrew Howroyd, Aug 21 2019

A306715 Number of graphical necklaces with n vertices and distinct rotations.

Original entry on oeis.org

1, 0, 2, 12, 204, 5372, 299592, 33546240, 7635496960, 3518433853392, 3275345183542176, 6148914685509544960, 23248573454127484128960, 176848577040728399988915648, 2704321280486889389857342715776, 83076749736557240903566436660674560
Offset: 1

Views

Author

Gus Wiseman, Mar 05 2019

Keywords

Comments

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. A graphical necklace is a simple graph that is minimal among all n rotations of the vertices.

Links

Crossrefs

Cf. A000088, A001037, A006125, A059966, A060223, A086675, A192332 (graphical necklaces), A306669, A323861, A323865, A323866, A323871, A324461 (distinct rotations), A324513.

Programs

  • Mathematica
    rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],With[{rots=Table[Nest[rotgra[#,n]&,#,j],{j,n}]},UnsameQ@@rots&&#==First[Sort[rots]]]&]],{n,5}]
  • PARI
    a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d))/n)} \\ Andrew Howroyd, Aug 15 2019

Formula

a(n > 0) = A324461(n)/n.
a(n) = (1/n)*Sum_{d|n} mu(d)*2^(n*(n/d-1)/2 + n*floor(d/2)/d) for n > 0. - Andrew Howroyd, Aug 15 2019

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 15 2019
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