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A323866 Number of aperiodic toroidal necklaces of positive integers summing to n. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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.
LINKS
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
EXAMPLE
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]
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}]
PROG
(GAP) List([0..30], A323866); # See A323861 for code; Andrew Howroyd, Aug 21 2019
CROSSREFS
Sequence in context: A025088 A279784 A198301 * A082740 A010067 A341710
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)