A323859 Number of binary toroidal necklaces of size n.

1, 2, 6, 8, 19, 16, 56, 40, 152, 184, 432, 376, 2132, 1264, 4728, 8768, 20688, 15424, 87656, 55192, 315128, 399520, 762984, 729448, 5595408, 4026576, 10325712, 19884504, 57527804, 37025584, 286340544, 138547336, 805335364, 1041204704, 2021176512, 3926827328

Offset: 0

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn

Comments

The 1-dimensional (necklace) case is A000031.

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.

Examples

			Inequivalent representatives of the a(4) = 19 binary toroidal necklaces:
  [0 0 0 0] [0 0 0 1] [0 0 1 1] [0 1 0 1] [0 1 1 1] [1 1 1 1]
.
  [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]
.
  [0] [0] [0] [0] [0] [1]
  [0] [0] [0] [1] [1] [1]
  [0] [0] [1] [0] [1] [1]
  [0] [1] [1] [1] [1] [1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Crossrefs

Cf. A000031, A008965, A179043, A184271, A323351.

Cf. A323858, A323859, A323865, A323870.

Programs

Mathematica

matcyc[m_]:=Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[If[n==0,1,Length[Union[First/@matcyc/@Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n])]]],{n,0,10}]

PARI

U(n, m, k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * k^(n*m/lcm(c, d))))
a(n) = if(n<1, n==0, sumdiv(n, d, U(n/d, d, 2))) \\ Andrew Howroyd, Jan 24 2023

Formula

a(n) = (1/n) * Sum_{d|n} Sum_{e|d, f|(n/d)} phi(e) * phi(f) * 2^(n/lcm(d,n/d)). [Ethier]