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A323871 Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers. 11
1, 2, 8, 53, 216, 3112, 13512, 272844, 2362412, 40898808, 295024104, 14045779864, 81055130520, 3040383692328, 61408850927280, 1661142087743940, 15337737297545400, 1128511554416582908, 9768588138876674856, 803306338873264137240, 15452347618762680730384 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

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.

EXAMPLE

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]

MATHEMATICA

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}]

PROG

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

CROSSREFS

Cf. A000670, A000740, A008965, A060223.

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

Sequence in context: A191507 A191602 A145157 * A183945 A193651 A195979

Adjacent sequences:  A323868 A323869 A323870 * A323872 A323873 A323874

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 04 2019

EXTENSIONS

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

STATUS

approved

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Last modified May 8 23:31 EDT 2021. Contains 343683 sequences. (Running on oeis4.)