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
Cf.
A000088,
A000740,
A003436,
A006125,
A027375,
A192314,
A192332,
A306669,
A306715,
A323860,
A323864,
A323867,
A324462 (covering case),
A324463,
A324464.
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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}]
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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
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
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.
-
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
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
The a(4) = 16 aperiodic permutations:
(1243) (1324) (1342) (1423)
(2134) (2314) (2413) (2431)
(3124) (3142) (3241) (3421)
(4132) (4213) (4231) (4312)
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Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]],{n,6}]
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a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ 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
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}]
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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
Showing 1-4 of 4 results.
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