A192332 -id:A192332 - cp's OEIS frontend

A000939 Number of inequivalent n-gons.

1, 1, 1, 2, 4, 14, 54, 332, 2246, 18264, 164950, 1664354, 18423144, 222406776, 2905943328, 40865005494, 615376173184, 9880209206458, 168483518571798, 3041127561315224, 57926238289970076, 1161157777643184900, 24434798429947993054, 538583682082245127336

Offset: 1

N. J. A. Sloane

Here two n-gons are said to be equivalent if they differ in starting point, orientation, or by a rotation (but not by a reflection - for that see A000940).

Number of cycle necklaces on n vertices, defined as equivalence classes of (labeled, undirected) Hamiltonian cycles under rotation of the vertices. The path version is A275527. - Gus Wiseman, Mar 02 2019

			Possibilities for n-gons without distinguished vertex can be encoded as permutation classes of vertices, two permutations being equivalent if they can be obtained from each other by circular rotation, translation mod n or complement to n+1.
n=3: 123.
n=4: 1234, 1243.
n=5: 12345, 12354, 12453, 13524.
n=6: 123456, 123465, 123564, 123645, 123654, 124365, 124635, 124653, 125364, 125463, 125634, 126435, 126453, 135264.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 1..450 (first 100 terms by T. D. Noe)
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]
Samuel Herman and Eirini Poimenidou, Orbits of Hamiltonian Paths and Cycles in Complete Graphs, arXiv:1905.04785 [math.CO], 2019.
A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Wikipedia, Hamiltonian path.
Wikipedia, Polygon.
Gus Wiseman, Inequivalent representatives of the a(6) = 14 cycle necklaces.
Gus Wiseman, Inequivalent representatives of the a(7) = 54 n-gons.

Cf. A000940. Bisections give A094154, A094155.
For star polygons see A231091.
Cf. A000031, A002619, A002866, A006125, A008965, A059966, A060223, A192332, A275527, A323858, A323870, A324461.

with(numtheory):
# for n odd:
Ed:= proc(n) local t1, d; t1:=0; for d from 1 to n do
       if n mod d = 0 then t1:= t1+phi(n/d)^2*d!*(n/d)^d fi od:
       t1/(2*n^2)
     end:
# for n even:
Ee:= proc(n) local t1, d; t1:= 2^(n/2)*(n/2)*(n/2)!; for d
       from 1 to n do if n mod d = 0 then t1:= t1+
       phi(n/d)^2*d!*(n/d)^d; fi od: t1/(2*n^2)
     end:
A000939:= n-> if n mod 2 = 0 then ceil(Ee(n)) else ceil(Ed(n)); fi:
seq(A000939(n), n=1..25);

a[n_] := (t = If[OddQ[n], 0, 2^(n/2)*(n/2)*(n/2)!]; Do[If[Mod[n, d]==0, t = t+EulerPhi[n/d]^2*d!*(n/d)^d], {d, 1, n}]; t/(2*n^2)); a[1] := 1; a[2] := 1; Print[a /@ Range[1, 450]] (* Jean-François Alcover, May 19 2011, after Maple prog. *)
rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,8}] (* Gus Wiseman, Mar 02 2019 *)

a(n)={if(n<3, n>=0, (if(n%2, 0, (n/2-1)!*2^(n/2-2)) + sumdiv(n, d, eulerphi(n/d)^2 * d! * (n/d)^d)/n^2)/2)} \\ Andrew Howroyd, Aug 17 2019

For formula see Maple lines.
a(2*n + 1) =  A002619(2*n + 1)/2 for n > 0; a(2*n) = (A002619(2*n) + A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 17 2019
a(n) ~ sqrt(2*Pi)/2 * n^(n-3/2) / e^n. - Ludovic Schwob, Nov 03 2022

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

Added a(1) = 1 and a(2) = 1 by Gus Wiseman, Mar 02 2019

A061417 Number of permutations up to cyclic rotations; permutation siteswap necklaces.

Original entry on oeis.org

1, 2, 4, 10, 28, 136, 726, 5100, 40362, 363288, 3628810, 39921044, 479001612, 6227066928, 87178295296, 1307675013928, 20922789888016, 355687438476444, 6402373705728018, 121645100594641896, 2432902008177690360, 51090942175425331320, 1124000727777607680022

Offset: 1

Antti Karttunen, May 02 2001

If permutations are converted to (i,p(i)) permutation arrays, then this automorphism is obtained by their "SW-NE diagonal toroidal shifts" (see Matthias Engelhardt's Java program in A006841), while the Maple procedure below converts each permutation to a siteswap pattern (used in juggling), rotates it by one digit and converts the resulting new (or same) siteswap pattern back to a permutation.
When the subset of permutations listed by A064640 are subjected to the same automorphism one gets A002995.
The number of conjugacy classes of the symmetric group of degree n when conjugating only with the cyclic permutation group of degree n. - Attila Egri-Nagy, Aug 15 2014
Also the number of equivalence classes of permutations of {1...n} under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514). - Gus Wiseman, Mar 04 2019

			If I have a five-element permutation like 25431, in cycle notation (1 2 5)(3 4), I mark the numbers 1-5 clockwise onto a circle and draw directed edges from 1 to 2, from 2 to 5, from 5 to 1 and a double-way edge between 3 and 4. All the 5-element permutations that produce some rotation (discarding the labels of the nodes) of that chord diagram belong to the same equivalence class with 25431. The sequence gives the count of such equivalence classes.

Cf. A006841, A060495. For other Maple procedures, see A060501 (Perm2SiteSwap2), A057502 (CountCycles), A057509 (rotateL), A060125 (PermRank3R and permul).
A061417[p] = A061860[p] = (p-1)!+(p-1) for all prime p's.
A064636 (derangements-the same automorphism).
A061417[n] = A064649[n]/n.
Cf. A000031, A000939, A002995, A008965, A060223, A064640, A086675 (digraphical necklaces), A179043, A192332, A275527 (path necklaces), A323858, A323859, A323870, A324513, A324514 (aperiodic permutations).

List([1..10],n->Size( OrbitsDomain( CyclicGroup(IsPermGroup,n), SymmetricGroup( IsPermGroup,n),\^))); # Attila Egri-Nagy, Aug 15 2014

a061417 = sum . a047917_row  -- Reinhard Zumkeller, Mar 19 2014

Algebraic formula: with(numtheory); SSRPCC := proc(n) local d,s; s := 0; for d in divisors(n) do s := s + phi(n/d)*((n/d)^d)*(d!); od; RETURN(s/n); end;
Empirically: with(group); SiteSwapRotationPermutationCycleCounts := proc(upto_n) local b,u,n,a,r; a := []; for n from 1 to upto_n do b := []; u := n!; for r from 0 to u-1 do b := [op(b),1+PermRank3R(SiteSwap2Perm1(rotateL(Perm2SiteSwap2(PermUnrank3Rfix(n,r)))))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
PermUnrank3Rfixaux := proc(n,r,p) local s; if(0 = n) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Rfixaux(n-1, r-(s*((n-1)!)), permul(p,[[n,n-s]]))); fi; end;
PermUnrank3Rfix := (n,r) -> convert(PermUnrank3Rfixaux(n,r,[]),'permlist',n);
SiteSwap2Perm1 := proc(s) local e,n,i,a; n := nops(s); a := []; for i from 1 to n do e := ((i+s[i]) mod n); if(0 = e) then e := n; fi; a := [op(a),e]; od; RETURN(convert(invperm(convert(a,'disjcyc')),'permlist',n)); end;

a[n_] := (1/n)*Sum[ EulerPhi[n/d]*(n/d)^d*d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Oct 09 2012, from formula *)
Table[Length[Select[Permutations[Range[n]],#==First[Sort[NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]]]&]],{n,8}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*sumdiv(n, d, eulerphi(n/d)*(n/d)^d*d!); \\ Indranil Ghosh, Apr 10 2017

from sympy import divisors, factorial, totient
def a(n):
    return sum(totient(n//d)*(n//d)**d*factorial(d) for d in divisors(n))//n
print([a(n) for n in range(1, 22)]) # Indranil Ghosh, Apr 10 2017

a(n) = (1/n)*Sum_{d|n} phi(n/d)*((n/d)^d)*(d!).

A324461 Number of simple graphs with n vertices and distinct rotations.

Original entry on oeis.org

1, 1, 0, 6, 48, 1020, 32232, 2097144, 268369920, 68719472640, 35184338533920, 36028797018963936, 73786976226114539520, 302231454903657293676480, 2475880078570197599844819072, 40564819207303340847860140736640, 1329227995784915854457062986570792960

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 48 graphs with distinct rotations.

Cf. A000088, A000740, A003436, A006125, A027375, A192314, A192332, A306669, A306715, A323860, A323864, A323867, A324462 (covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,0,5}]

a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d)))} \\ Andrew Howroyd, Aug 15 2019

from sympy import mobius, divisors
def A324461(n): return sum(mobius(m:=n//d)<<(n*(d-1)>>1)+d*(m>>1) for d in divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Jul 03 2024

a(n > 0) = A306715(n) * n.

a(n) = Sum_{d|n} mu(d)*2^(n*(n/d-1)/2 + n*floor(d/2)/d) for n > 0. - Andrew Howroyd, Aug 15 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 15 2019

A275527 Number of distinct classes of permutations of length n under reversal and complement to n+1.

Original entry on oeis.org

1, 1, 1, 4, 12, 64, 360, 2544, 20160, 181632

Offset: 1

Olivier Gérard, Jul 31 2016

Let us consider two permutations to be equivalent if they can be obtained from each other by cyclic rotation (12345->(23451,34512,45123,51234) or n+1-complement (31254->35412), or a combination of those two transformations (they commute with each other). a(n) is the number of classes.
We obtain the same number of classes if the transformations are (addition of a constant modulo n and reversal (12345->54321)) but not the same set of representatives.
It seems probable that a(2n+1) = (2n)!/2
This sequence may be related to A113247 (and A113248) as they share a common dissection 1, 4, 64, 2544, 181632. The fact that they count permutation classes for the major index is a further indication.
Number of path necklaces, defined as equivalence classes of (labeled, undirected) Hamiltonian paths under rotation of the vertices. The cycle version is A000939. - Gus Wiseman, Mar 02 2019

			Examples of permutation representatives. The representative is chosen to be the first of the class in lexicographic order.
n=4 case addition mod n and reversal
1234, 1243, 1324, 1423.
n=4 case rotation and complement
1234, 1243, 1324, 1342.
.
n=5 case addition mod n and reversal
12345, 12354, 12435, 12453, 12534, 13245, 13425, 13452, 13524, 14235, 14523, 15234.
n=5 case rotation and complement
12345, 12354, 12435, 12453, 12534, 13245, 13425, 13452, 13524, 14235, 14325, 14352.

Wikipedia, Hamiltonian path.
Wikipedia, Necklace (combinatorics).
Gus Wiseman, Inequivalent representatives of the a(5) = 12 path necklaces.
Gus Wiseman, Inequivalent representatives of the a(6) = 54 path necklaces.

Cf. A000939, A000940, A002619, A089066, A262480 (other symmetry classes of permutations).
Cf. A193651 (inspiration for a(2n)).
Cf. A000031, A006125, A008965, A059966, A060223, A192332, A323858, A323870, A324461.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,8}] (* Gus Wiseman, Mar 02 2019 *)

(Conjecture). If n odd a(n)=((n - 1))!/2. If n even a(n)= 1/2 (n - 2)!! (1 + ( n - 1)!!).

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

Original entry on oeis.org

1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976

Offset: 0

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

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

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..57
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

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.

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 *)

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

More terms from Wouter Meeussen, Jul 29 2003

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

A324513 Number of aperiodic cycle necklaces with n vertices.

Original entry on oeis.org

1, 0, 0, 0, 2, 7, 51, 300, 2238, 18028, 164945, 1662067, 18423138, 222380433, 2905942904, 40864642560, 615376173176, 9880203467184, 168483518571789, 3041127459127222, 57926238289894992, 1161157775616335125, 24434798429947993043, 538583682037962702384

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

We define an aperiodic cycle necklace to be an equivalence class of (labeled, undirected) Hamiltonian cycles under rotation of the vertices such that all n of these rotations are distinct.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(6) = 7 aperiodic cycle necklaces.
Gus Wiseman, Inequivalent representatives of the a(7) = 51 aperiodic cycle necklaces.

Cf. A000740, A000939, A001037 (binary Lyndon words), A008965, A059966 (Lyndon compositions), A060223 (normal Lyndon words), A061417, A064852 (if cycle is oriented), A086675, A192332, A275527, A323866 (aperiodic toroidal arrays), A323871.

Cf. A306669, A324461, A324462, A324512, A324514.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&&UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]

a(n)={if(n<3, n==0||n==1, (if(n%2, 0, -(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2))/2)} \\ Andrew Howroyd, Aug 19 2019

a(n) = A324512(n)/n.

a(2*n+1) = A064852(2*n+1)/2 for n > 0; a(2*n) = (A064852(2*n) - A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 16 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A324462 Number of simple graphs covering n vertices with distinct rotations.

Original entry on oeis.org

1, 0, 0, 3, 28, 765, 26958, 1887277, 252458904, 66376420155, 34508978662350, 35645504882731557, 73356937843604425644, 301275024444053951967585, 2471655539736990372520379226, 40527712706903544100966076156895, 1328579255614092949957261201822704816

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. It is covering if there are no isolated vertices. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations.
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations, radial embedding.

Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]]&]],{n,0,5}]

a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2)))))} \\ Andrew Howroyd, Aug 19 2019

a(n) = Sum{d|n} mu(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ) for n > 0. - Andrew Howroyd, Aug 19 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

A324514 Number of aperiodic permutations of {1..n}.

Original entry on oeis.org

1, 0, 3, 16, 115, 660, 5033, 39936, 362718, 3624920, 39916789, 478953648, 6227020787, 87177645996, 1307674338105, 20922779566080, 355687428095983, 6402373519409856, 121645100408831981, 2432902004460734000, 51090942171698415483, 1124000727695858073380

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is defined to be aperiodic if every cyclic rotation of {1..n} acts on the cycle decomposition to produce a different digraph.

			The a(4) = 16 aperiodic permutations:
  (1243) (1324) (1342) (1423)
  (2134) (2314) (2413) (2431)
  (3124) (3142) (3241) (3421)
  (4132) (4213) (4231) (4312)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Cycle decompositions of the a(4) = 16 aperiodic permutations.

Cf. A000740, A000939, A027375, A061417, A079128, A086675, A192332, A323860, A323867, A323869.

Cf. A306669, A324461, A324462, A324512, A324513, A324514.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]],{n,6}]

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A306669(n) * n.

a(n) = Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A324463 Number of graphical necklaces covering n vertices.

Original entry on oeis.org

1, 0, 1, 2, 15, 156, 4665, 269618, 31573327, 7375159140, 3450904512841, 3240500443884718, 6113078165054644451, 23175001880311842459108, 176546824267008236554238517, 2701847513793569606737940203894, 83036203475880811677609125194805687

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. Covering means there are no isolated vertices. These are a kind of partially labeled graphs.

			Inequivalent representatives of the a(2) = 1 through a(4) = 15 graphical necklaces:
  {{12}}  {{12}{13}}      {{12}{34}}
          {{12}{13}{23}}  {{13}{24}}
                          {{12}{13}{14}}
                          {{12}{13}{24}}
                          {{12}{13}{34}}
                          {{12}{14}{23}}
                          {{12}{24}{34}}
                          {{12}{13}{14}{23}}
                          {{12}{13}{14}{24}}
                          {{12}{13}{14}{34}}
                          {{12}{13}{24}{34}}
                          {{12}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}}
                          {{12}{13}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 15 covering graphical necklaces, radial embedding.
Gus Wiseman, The a(5) = 156 covering graphical necklaces.

Cf. A000031, A002494, A006129, A008965, A184271, A192332 (non-covering case), A323858, A323859, A323870, A324461, A324462, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]]&]],{n,0,5}]

a(n)={if(n<1, n==0, sumdiv(n, d, eulerphi(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2))))/n)} \\ Andrew Howroyd, Aug 19 2019

a(n) = (1/n)*Sum{d|n} phi(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ). - Andrew Howroyd, Aug 19 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

A324464 Number of connected graphical necklaces with n vertices.

Original entry on oeis.org

1, 0, 1, 2, 13, 148, 4530, 266614, 31451264, 7366255436, 3449652145180, 3240150686268514, 6112883022923529310, 23174784819204929919428, 176546343645071836902594288, 2701845395848698682311893154024, 83036184895986451215378727412638816, 5122922885438069578928905234650082410736

Offset: 0

Gus Wiseman, Mar 01 2019

nonn

A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs.

			Inequivalent representatives of the a(2) = 1 through a(4) = 13 graphical necklaces:
  {{12}}  {{12}{13}}      {{12}{13}{14}}
          {{12}{13}{23}}  {{12}{13}{24}}
                          {{12}{13}{34}}
                          {{12}{14}{23}}
                          {{12}{24}{34}}
                          {{12}{13}{14}{23}}
                          {{12}{13}{14}{24}}
                          {{12}{13}{14}{34}}
                          {{12}{13}{24}{34}}
                          {{12}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}}
                          {{12}{13}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 13 connected graphical necklaces.
Gus Wiseman, The a(5) = 148 connected graphical necklaces.

Cf. A000031, A000939, A001187, A006125, A006129, A008965, A184271, A192332, A275527, A323858, A323859, A323870, A324461, A324462, A324463.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[csm[#]]<=1,#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]]&]],{n,0,5}]

\\ B(n,d) is graphs on n*d points invariant under 1/d rotation.
B(n,d)={2^(n*(n-1)*d/2 + n*(d\2))}
D(n,d)={my(v=vector(n, i, B(i,d)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); sumdiv(n, e, eulerphi(d*e) * moebius(e) * u[n/e] * e^(n/e-1))}
a(n)={if(n<=1, n==0, sumdiv(n, d, D(n/d,d))/n)} \\ Andrew Howroyd, Jan 24 2023

Terms a(7) and beyond from Andrew Howroyd, Jan 24 2023

cp's OEIS Frontend

A000939 Number of inequivalent n-gons.

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A061417 Number of permutations up to cyclic rotations; permutation siteswap necklaces.

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A324461 Number of simple graphs with n vertices and distinct rotations.

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A275527 Number of distinct classes of permutations of length n under reversal and complement to n+1.

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A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

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A324513 Number of aperiodic cycle necklaces with n vertices.

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