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A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows. 8
1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also the number of digraphical necklaces with n vertices. A digraphical necklace is defined to be a directed graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of directed graphs under rotation of the vertices. These are a kind of partially labeled digraphs. - Gus Wiseman, Mar 04 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..57

FORMULA

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n^2/d) for n > 0, a(0) = 1.

EXAMPLE

From Gus Wiseman, Mar 04 2019: (Start)

Inequivalent representatives of the a(2) = 10 digraphical necklace edge-sets:

  {}

  {(1,1)}

  {(1,2)}

  {(1,1),(1,2)}

  {(1,1),(2,1)}

  {(1,1),(2,2)}

  {(1,2),(2,1)}

  {(1,1),(1,2),(2,1)}

  {(1,1),(1,2),(2,2)}

  {(1,1),(1,2),(2,1),(2,2)}

(End)

MATHEMATICA

Table[Fold[ #1+EulerPhi[ #2] 2^(n^2 /#2)&, 0, Divisors[n]]/n, {n, 16}]

(* second program *)

rotdigra[g_, m_]:=Sort[g/.k_Integer:>If[k==m, 1, k+1]];

Table[Length[Select[Subsets[Tuples[Range[n], 2]], #=={}||#==First[Sort[Table[Nest[rotdigra[#, n]&, #, j], {j, n}]]]&]], {n, 0, 4}] (* Gus Wiseman, Mar 04 2019 *)

CROSSREFS

Cf. A000031 (binary necklaces), A000939 (cycle necklaces), A008965, A060690, A061417 (permutation necklaces), A184271, A192332 (graphical necklaces), A275527 (path necklaces), A323858 (toroidal necklaces), A323870.

Cf. A324463, A324464, A324513, A324514.

Sequence in context: A132341 A069994 A063573 * A319607 A057119 A226563

Adjacent sequences:  A086672 A086673 A086674 * A086676 A086677 A086678

KEYWORD

nonn,easy

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 27 2003

EXTENSIONS

More terms from Wouter Meeussen, Jul 29 2003

a(0)=1 prepended by Gus Wiseman, Mar 04 2019

STATUS

approved

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Last modified September 23 07:00 EDT 2021. Contains 347609 sequences. (Running on oeis4.)