A323864 -id:A323864 - cp's OEIS frontend

A323865 Number of aperiodic binary toroidal necklaces of size n.

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

A324461 Number of simple graphs with n vertices and distinct rotations.

Original entry on oeis.org

1, 1, 0, 6, 48, 1020, 32232, 2097144, 268369920, 68719472640, 35184338533920, 36028797018963936, 73786976226114539520, 302231454903657293676480, 2475880078570197599844819072, 40564819207303340847860140736640, 1329227995784915854457062986570792960

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

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. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 48 graphs with distinct rotations.

Cf. A000088, A000740, A003436, A006125, A027375, A192314, A192332, A306669, A306715, A323860, A323864, A323867, A324462 (covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,0,5}]

a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d)))} \\ Andrew Howroyd, Aug 15 2019

from sympy import mobius, divisors
def A324461(n): return sum(mobius(m:=n//d)<<(n*(d-1)>>1)+d*(m>>1) for d in divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Jul 03 2024

a(n > 0) = A306715(n) * n.

a(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

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

A323863 Number of n X n aperiodic binary arrays.

Original entry on oeis.org

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset: 0

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(2) = 8 arrays are:
  [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 = 0..50

Cf. A000031, A000740, A027375, A179043, A265627, A323351.

Cf. A323860, A323862, A323864, A323865, A323867, A323869.

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

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

a(5) from Robert Israel, Feb 04 2019
a(6)-a(7) from Giovanni Resta, Feb 05 2019
Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

A324462 Number of simple graphs covering n vertices with distinct rotations.

Original entry on oeis.org

1, 0, 0, 3, 28, 765, 26958, 1887277, 252458904, 66376420155, 34508978662350, 35645504882731557, 73356937843604425644, 301275024444053951967585, 2471655539736990372520379226, 40527712706903544100966076156895, 1328579255614092949957261201822704816

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

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. It is covering if there are no isolated vertices. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations.
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations, radial embedding.

Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]]&]],{n,0,5}]

a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2)))))} \\ Andrew Howroyd, Aug 19 2019

a(n) = Sum{d|n} mu(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ) for n > 0. - Andrew Howroyd, Aug 19 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

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

Gus Wiseman, Feb 04 2019

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

			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

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

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

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}]

A(n,k) = {sumdiv(n, d, sumdiv(k,e, moebius(d) * moebius(e) * 2^((n/d) * (k/e))))} \\ Andrew Howroyd, Jan 19 2023

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

cp's OEIS Frontend

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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A324461 Number of simple graphs with n vertices and distinct rotations.

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

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A323863 Number of n X n aperiodic binary arrays.

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A324462 Number of simple graphs covering n vertices with distinct rotations.

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