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

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

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn

Comments

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.

Examples

			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]
		

Links

• 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.

Crossrefs

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

Cf. A323858, A323861, A323865, A323871.

Programs

GAP

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

Mathematica

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}]

Extensions

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