A086675 -id:A086675 - cp's OEIS frontend

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

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!).

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))).

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

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

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

A306715 Number of graphical necklaces with n vertices and distinct rotations.

Original entry on oeis.org

1, 0, 2, 12, 204, 5372, 299592, 33546240, 7635496960, 3518433853392, 3275345183542176, 6148914685509544960, 23248573454127484128960, 176848577040728399988915648, 2704321280486889389857342715776, 83076749736557240903566436660674560

Offset: 1

Gus Wiseman, Mar 05 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. A graphical necklace is a simple graph that is minimal among all n rotations of the vertices.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Gus Wiseman, Inequivalent representatives of the a(4) = 12 graphical necklaces with distinct rotations.

Cf. A000088, A001037, A006125, A059966, A060223, A086675, A192332 (graphical necklaces), A306669, A323861, A323865, A323866, A323871, A324461 (distinct rotations), A324513.

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

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

a(n > 0) = A324461(n)/n.

a(n) = (1/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

cp's OEIS Frontend

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

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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.

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A324513 Number of aperiodic cycle necklaces with n vertices.

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

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A306669 Number of aperiodic permutation necklaces of weight n.

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A306715 Number of graphical necklaces with n vertices and distinct rotations.

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A086683 Number of n X n {-1,0,1} matrices modulo cyclic permutations of the rows.