A324461 -id:A324461 - 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

A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.

Original entry on oeis.org

1, 2, 4, 22, 208, 5560, 299600, 33562696, 7635498336, 3518440564544, 3275345183542208, 6148914696963883712, 23248573454127484129024, 176848577040808821410837120, 2704321280486889389864215362560, 83076749736557243209409446411255936, 5124252113632955685095523500148980125696, 634332307869315502692705867068871886072665600

Offset: 1

N. J. A. Sloane, Jun 28 2011

nonn

Suggested by A192314.

Also the number of graphical necklaces with n vertices. We define a graphical necklace to be 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. - Gus Wiseman, Mar 04 2019

			From _Gus Wiseman_, Mar 04 2019: (Start)
Inequivalent representatives of the a(1) = 1 through a(4) = 22 graphical necklace edge-sets:
  {}  {}      {}              {}
      {{12}}  {{12}}          {{12}}
              {{12}{13}}      {{13}}
              {{12}{13}{23}}  {{12}{13}}
                              {{12}{14}}
                              {{12}{24}}
                              {{12}{34}}
                              {{13}{24}}
                              {{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}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..80
Gus Wiseman, Inequivalent representatives of the a(5) = 22 graphical necklaces.
Gus Wiseman, Inequivalent representatives of the a(5) = 208 graphical necklaces.

Cf. A192314, A191563 (orbits under dihedral group).

Cf. A000031, A000939 (cycle necklaces), A008965, A059966, A060223, A061417, A086675 (digraph version), A184271, A275527, A323858, A324461, A324463, A324464.

with(numtheory);
f:=proc(n) local t0, t1, d; t0:=0; t1:=divisors(n);
for d in t1 do
if d mod 2 = 0 then t0:=t0+phi(d)*2^(n^2/(2*d))
else t0:=t0+phi(d)*2^(n*(n-1)/(2*d)); fi; od; t0/n; end;
[seq(f(n), n=1..30)];

Table[ 1/n* Plus @@ Map[Function[d, EulerPhi[d]*2^((n*(n - Mod[d, 2])/2)/d)], Divisors[n]], {n, 1, 20}]  (* Olivier Gérard, Aug 27 2011 *)
rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,0,5}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = sumdiv(n, d, if (d%2, eulerphi(d)*2^(n*(n-1)/(2*d)), eulerphi(d)*2^(n^2/(2*d))))/n; \\ Michel Marcus, Mar 08 2019

a(n) = (1/n)*(Sum_{d|n, d odd} phi(d)*2^(n*(n-1)/(2*d)) + Sum_{d|n, d even} phi(d)*2^(n^2/(2*d))).

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)!!).

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

A306669 Number of aperiodic permutation necklaces of weight n.

Original entry on oeis.org

1, 0, 1, 4, 23, 110, 719, 4992, 40302, 362492, 3628799, 39912804, 479001599, 6226974714, 87178289207, 1307673722880, 20922789887999, 355687417744992, 6402373705727999, 121645100223036700, 2432902008176115023, 51090942167993548790, 1124000727777607679999

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations 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).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(5) = 23 aperiodic necklace permutations.

Cf. A000031, A000740, A000939, A001037, A059966, A060223, A061417, A086675, A323861, A323865, A323866, A323871.

Cf. A324461, A324512, A324513, A324514.

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

a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A324514(n)/n.

a(n) = (1/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

A324512 Number of aperiodic n-gons.

Original entry on oeis.org

1, 0, 0, 0, 10, 42, 357, 2400, 20142, 180280, 1814395, 19944804, 239500794, 3113326062, 43589143560, 653834280960, 10461394943992, 177843662409312, 3201186852863991, 60822549182544440, 1216451004087794832, 25545471063559372750, 562000363888803839989

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

We define an n-gon to be aperiodic if all n rotations of its vertex set act on the edge set to give distinct n-gons. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

			The a(5) = 10 aperiodic polygon edge sets:
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,2},{1,3},{2,5},{3,4},{4,5}}
  {{1,2},{1,4},{2,3},{3,5},{4,5}}
  {{1,2},{1,4},{2,5},{3,4},{3,5}}
  {{1,2},{1,5},{2,4},{3,4},{3,5}}
  {{1,3},{1,4},{2,3},{2,5},{4,5}}
  {{1,3},{1,5},{2,3},{2,4},{4,5}}
  {{1,3},{1,5},{2,4},{2,5},{3,4}}
  {{1,4},{1,5},{2,3},{2,4},{3,5}}
  {{1,4},{1,5},{2,3},{2,5},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, The a(5) = 10 aperiodic polygons.
Gus Wiseman, The a(6) = 42 aperiodic polygons.

Cf. A000740, A008965, A027375, A059966, A060223, A192332, A275527, A323860, A323867, A323869, A324461, A324462, A324513, 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]]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]

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

a(n) = n * A324513(n).

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

cp's OEIS Frontend

A000939 Number of inequivalent n-gons.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A324513 Number of aperiodic cycle necklaces with n vertices.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A324463 Number of graphical necklaces covering n vertices.

Views

Author