A323861 -id:A323861 - cp's OEIS frontend

A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).

2, 3, 3, 4, 7, 4, 6, 14, 14, 6, 8, 40, 64, 40, 8, 14, 108, 352, 352, 108, 14, 20, 362, 2192, 4156, 2192, 362, 20, 36, 1182, 14624, 52488, 52488, 14624, 1182, 36, 60, 4150, 99880, 699600, 1342208, 699600, 99880, 4150, 60, 108, 14602, 699252, 9587580, 35792568

Offset: 1

R. H. Hardin, Jan 10 2011

This is a 2-dimensional generalization of binary necklaces (A000031). A toroidal array or necklace can be defined either as an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns, or as a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns. - Gus Wiseman, Feb 04 2019

			      1     2        3           4            5             6              7
----------------------------------------------------------------------------
1:    2     3        4           6            8            14             20
2:    3     7       14          40          108           362           1182
3:    4    14       64         352         2192         14624          99880
4:    6    40      352        4156        52488        699600        9587580
5:    8   108     2192       52488      1342208      35792568      981706832
6:   14   362    14624      699600     35792568    1908897152   104715443852
7:   20  1182    99880     9587580    981706832  104715443852 11488774559744
8:   36  4150   699252   134223976  27487816992 5864063066500
9:   60 14602  4971184  1908881900 781874936816
10: 108 52588 35792568 27487869472
From _Gus Wiseman_, Feb 04 2019: (Start)
Inequivalent representatives of the T(2,3) = 14 toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
  [0 0 0] [0 0 1] [0 1 1] [1 1 1] [0 0 1] [0 1 0] [0 1 1]
.
  [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
  [1 0 1] [1 1 0] [1 1 1] [0 1 1] [1 0 1] [1 1 1] [1 1 1]
(End)

Alois P. Heinz, Antidiagonals n = 1..100, flattened (first 95 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.
Wikipedia, Pólya enumeration theorem

Columns 1-8 are A000031, A184264, A184265, A184266, A184267, A184268, A184269, A184270.
Main diagonal is A179043.
Cf. A001037 (binary Lyndon words), A008965, A323858, A323859 (binary toroidal necklaces of size n), A323861 (aperiodic version), A323865, A323870 (normal toroidal necklaces), A323872.

a[n_, k_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[k, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n k/LCM[c, d]), 0], {d, 1, k}], 0], {c, 1, n}]/(n k)
(* second program *)
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],neckmatQ]],{n,8},{k,n-1}] (* Gus Wiseman, Feb 04 2019 *)

T(n,k) = (1/(nk))*Sum_{ c divides n } Sum_{ d divides k } phi(c)*phi(d)*2^(nk/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012

A179043 Number of n X n checkered tori.

Original entry on oeis.org

1, 2, 7, 64, 4156, 1342208, 1908897152, 11488774559744, 288230376353050816, 29850020237398264483840, 12676506002282327791964489728, 21970710674130840874443091905462272, 154866286100907105149651981766316633972736

Offset: 0

Rouben Rostamian (rostamian(AT)umbc.edu), Jun 25 2010

nonn

Consider an n X n checkerboard whose tiles are assigned colors 0 and 1, at random. There are 2^(n^2) such checkerboards. We identify the opposite edges of each checkerboard, thus making it into a (topological) torus. There are a(n) such (distinct) tori. It is possible to show that a(n) >= 2^(n^2)/n^2 for all n.
Main diagonal of A184271.
Main diagonal of Table 3: The number a(m, n) of toroidal m X n binary arrays, allowing rotation of the rows and/or the columns but not reflection, for m, n = 1, 2, ..., 8, at page 5 of Ethier. - Jonathan Vos Post, Jan 14 2013
This is a 2-dimensional generalization of binary necklaces (A000031). - Gus Wiseman, Feb 04 2019

			From _Gus Wiseman_, Feb 04 2019: (Start)
Inequivalent representatives of the a(2) = 7 checkered tori:
  [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
  [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..57
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Peter Kagey and William Keehn, Counting Tilings of the n X m Grid, Cylinder, and Torus, J. Int. Seq. (2024) Vol. 27, Art. No. 24.6.1. See p. 2.
Wikipedia, Pólya enumeration theorem

Cf. A184271 (n X k toroidal binary arrays).
Cf. A000031, A001037, A008965.
Cf. A323858, A323859, A323861, A323863, A323865, A323870, A323872.

a[n_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}]/n ^2

a(n) = (1/n^2)*Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)*2^(n^2/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012

Terms a(6) and a(7) from A184271
a(8)-a(12) from Stewart N. Ethier, Aug 24 2012
a(0)=1 prepended by Alois P. Heinz, Aug 20 2017

A323858 Number of toroidal necklaces of positive integers summing to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 31, 44, 90, 154, 296, 524, 1035, 1881, 3636, 6869, 13208, 25150, 48585, 93188, 180192, 347617, 673201, 1303259, 2529740, 4910708, 9549665, 18579828, 36192118, 70540863, 137620889, 268655549, 524873503, 1026068477, 2007178821, 3928564237

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A008965.

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. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(6) = 31 toroidal necklaces:
  6  15  24  33  114  123  132  222  1113  1122  1212  11112  111111
.
  1  2  3  11  11  12  12  111
  5  4  3  13  22  12  21  111
.
  1  1  1  2  11
  1  2  3  2  11
  4  3  2  2  11
.
  1  1  1
  1  1  2
  1  2  1
  3  2  2
.
  1
  1
  1
  1
  2
.
  1
  1
  1
  1
  1
  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A000670, A008965, A059966, A101509, A179043, A184271.

Cf. A323859, A323861, A323865, A323866, A323870.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Join@@Table[Select[ptnmats[k],neckmatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,10}]

U(n,m,k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * subst(k, x, x^lcm(c,d))^(n*m/lcm(c, d))));
a(n)={if(n < 1, n==0, sum(i=1, n, sum(j=1, n\i, polcoef(U(i, j, x/(1-x) + O(x*x^n)), n))))} \\ Andrew Howroyd, Aug 18 2019

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

A323865 Number of aperiodic binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

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.

			Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
  000001  000011  000101  000111  001011  001101  001111  010111  011111
.
  000  000  001  001  001  001  001  011  011
  001  011  010  011  101  110  111  101  111
.
  00  00  00  00  00  01  01  01  01
  00  01  01  01  11  01  01  10  11
  01  01  10  11  01  10  11  11  11
.
  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  1  1
  0  0  0  0  1  1  1  0  1
  0  0  1  1  0  1  1  1  1
  0  1  0  1  1  0  1  1  1
  1  1  1  1  1  1  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, A323871.

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]]}]];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]

a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

Original entry on oeis.org

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional case is A027375.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

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

First and last columns are A027375. Main diagonal is A323863.
Cf. A000740, A001037, A179043, A265627, A323351.
Cf. A323861, A323862, A323864, A323865, A323867, A323869.

# See A323861 for code.
for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019

A323867 Number of aperiodic arrays of positive integers summing to n.

Original entry on oeis.org

1, 1, 1, 5, 11, 33, 57, 157, 303, 683, 1358, 2974, 5932, 12560, 25328, 52400, 106256, 217875, 441278, 899955, 1822703, 3701401, 7491173, 15178253, 30691135, 62085846, 125435689, 253414326, 511547323, 1032427635, 2082551931, 4199956099, 8466869525, 17064777665

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional case is A000740.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(5) = 33 arrays:
  5  14  23  32  41  113  122  131  212  221  311  1112  1121  1211  2111
.
  1  2  3  4  11  11  12  21
  4  3  2  1  12  21  11  11
.
  1  1  1  2  2  3
  1  2  3  1  2  1
  3  2  1  2  1  1
.
  1  1  1  2
  1  1  2  1
  1  2  1  1
  2  1  1  1

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

Cf. A000670, A000740, A027375, A101509.

Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.

List([0..30], A323867); # See A323861 for code; Andrew Howroyd, Aug 21 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Union@@Table[Select[ptnmats[k],apermatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,15}]

Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019

A323871 Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 8, 53, 216, 3112, 13512, 272844, 2362412, 40898808, 295024104, 14045779864, 81055130520, 3040383692328, 61408850927280, 1661142087743940, 15337737297545400, 1128511554416582908, 9768588138876674856, 803306338873264137240, 15452347618762680730384

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (Lyndon word) case is A060223.

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.

			The a(3) = 8 aperiodic toroidal necklaces:
  [1 2 3] [1 3 2] [1 2 2] [1 1 2]
.
  [1] [1] [1] [1]
  [2] [3] [2] [1]
  [3] [2] [2] [2]

Andrew Howroyd, Table of n, a(n) for n = 1..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000670, A000740, A008965, A060223.

Cf. A323858, A323859, A323860, A323861, A323866, A323867, A323868, A323870.

List([1..30], A323871); # See A323861 for code; Andrew Howroyd, Aug 21 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
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[nrmmats[n],neckmatQ[#]&&apermatQ[#]&]],{n,6}]

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

A323866 Number of aperiodic toroidal necklaces of positive integers summing to n.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 18, 42, 72, 145, 262, 522, 960, 1879, 3531, 6831, 13013, 25148, 48177, 93186, 179507, 347509, 671955, 1303257, 2527162, 4910681, 9545176, 18579471, 36183505, 70540861, 137603801, 268655547, 524842088, 1026067205, 2007118657, 3928564113

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (Lyndon word) case is A059966.

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.

			Inequivalent representatives of the a(6) = 18 toroidal necklaces:
  [6] [1 5] [2 4] [1 1 4] [1 2 3] [1 3 2] [1 1 1 3] [1 1 2 2] [1 1 1 1 2]
.
  [1] [2] [1 1]
  [5] [4] [1 3]
.
  [1] [1] [1]
  [1] [2] [3]
  [4] [3] [2]
.
  [1] [1]
  [1] [1]
  [1] [2]
  [3] [2]
.
  [1]
  [1]
  [1]
  [1]
  [2]

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A000670, A000740, A001037, A008965, A059966, A060223, A185700.

Cf. A323858, A323861, A323865, A323871.

List([0..30], A323866); # See A323861 for code; Andrew Howroyd, Aug 21 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
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[If[n==0,1,Length[Union@@Table[Select[ptnmats[k],And[apermatQ[#],neckmatQ[#]]&],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]]],{n,0,10}]

Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019

A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 4, 24, 212, 1080, 18672, 94584, 2182752, 21261708, 408988080, 3245265144, 168549358368, 1053716696760, 42565371692592, 921132763909200, 26578273403903040, 260741534058271800, 20313207979498492344, 185603174638656822264, 16066126777465282744800, 324499299994016295338064

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(3) = 24 matrices:
  [123][132][213][312][231][321][122][211][112][221][121][212]
.
  [1][1][2][3][2][3][1][2][1][2][1][2]
  [2][3][1][1][3][2][2][1][1][2][2][1]
  [3][2][3][2][1][1][2][1][2][1][1][2]

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

Cf. A000670, A000740, A027375, A060223, A265627.

Cf. A323860, A323864, A323867, A323868, A323871.

List([1..30], A323869); # See A323861 for code; Andrew Howroyd, Aug 21 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[nrmmats[n],apermatQ]],{n,6}]

a(n) = n*A323871(n). - Andrew Howroyd, Aug 21 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

A306669 Number of aperiodic permutation necklaces of weight n.

Original entry on oeis.org

1, 0, 1, 4, 23, 110, 719, 4992, 40302, 362492, 3628799, 39912804, 479001599, 6226974714, 87178289207, 1307673722880, 20922789887999, 355687417744992, 6402373705727999, 121645100223036700, 2432902008176115023, 51090942167993548790, 1124000727777607679999

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(5) = 23 aperiodic necklace permutations.

Cf. A000031, A000740, A000939, A001037, A059966, A060223, A061417, A086675, A323861, A323865, A323866, A323871.

Cf. A324461, A324512, A324513, A324514.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]]/n,{n,6}]

a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A324514(n)/n.

a(n) = (1/n)*Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

cp's OEIS Frontend

A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).

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A179043 Number of n X n checkered tori.

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A323858 Number of toroidal necklaces of positive integers summing to n.

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A323865 Number of aperiodic binary toroidal necklaces of size n.

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A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

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A323867 Number of aperiodic arrays of positive integers summing to n.

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A323871 Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers.

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