A061417 -id:A061417 - cp's OEIS frontend

A064649 Row sums of the table A047916.

1, 4, 12, 40, 140, 816, 5082, 40800, 363258, 3632880, 39916910, 479052528, 6227020956, 87178936992, 1307674429440, 20922800222848, 355687428096272, 6402373892575992, 121645100408832342, 2432902011892837920

Offset: 1

Antti Karttunen, Oct 04 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..100

Also n*A061417[n]. Cf. A047918, A002619.

a064649 = sum . a047916_row  -- Reinhard Zumkeller, Mar 19 2014

A064649 := 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); end;

a[n_] := DivisorSum[n, EulerPhi[n/#]*(n/#)^#*#!&]; Array[a, 20] (* Jean-François Alcover, Mar 06 2016 *)

{ for (n=1, 100, a=0; v=divisors(n); for (i=1, length(v), d=v[i]; a+=eulerphi(n/d)*(n/d)^d*d!); write("b064649.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 21 2009

a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^d*d!); \\ Michel Marcus, Mar 06 2016

a(n) = Sum_{d|n} phi(n/d)*(n/d)^d*d!. - Michel Marcus, Mar 06 2016

A064636 Number of derangements up to cyclic rotations; permutation siteswap necklaces, with no fixed points (no "zero-throws", i.e., empty hands, if we use the mapping Perm2SiteSwap1 of A060495 and A060498).

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 55, 270, 1893, 14864, 133749, 1334970, 14687195, 176214852, 2290820923, 32071104006, 481066907653, 7697064251760, 130850098582189, 2355301661033970, 44750731672347273, 895014631193654828, 18795307257304746591, 413496759611120779902, 9510425471105377569963, 228250211305338670543432

Offset: 0

Antti Karttunen, Oct 02 2001

nonn

This sequence counts derangements (enumerated by A000166) up to the same automorphism as permutations (enumerated by A000142) are subjected to in A061417.

Juggling Information Service, Site Swap notation

with(numtheory); A064636 := proc(n) local d,k,s; s := 0; for d in divisors(n) do s := s + (1/n) * phi(n/d) * ( (((n/d)^d)*A000166(d)) + add((((n/d)^(d-k)) * (((n/d)-1)^k) * (A000166(d-k)*binomial(d,k))),k=1..d)); od; RETURN(s); end;

Unprotect[Power]; 0^0 = 1; a[n_] := (1/n) DivisorSum[n, EulerPhi[n/#]*Sum[ (n/#)^(# - k)*(n/# - 1)^k*#!*Gamma[# - k + 1, -1]/(E*k!*(# - k)!), {k, 0, #}]&] // FunctionExpand; a[0] = 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 06 2016 *)

a(n) = Sum_{d|n} (1/n) * Phi(n/d) * Sum_{k=0..d} [ ((n/d)^(d-k)) * (((n/d)-1)^k) * A008290(d, k) ]. (Note: this abbreviated formula supposes that 0^0 = 1. For a practical implementation, see the Maple procedure below.)

A242099 Number of conjugacy classes of the symmetric group S_n when conjugating by the dihedral group D_n.

Original entry on oeis.org

1, 2, 3, 8, 18, 84, 387, 2670, 20373, 182796, 1816325, 19973962, 239523846, 3113717784, 43589470208, 653840410004, 10461400104968, 177843770847822, 3201186945761289, 60822551319191028, 1216451005946790780, 25545471110008012860, 562000363929678643211

Offset: 1

Attila Egri-Nagy, Aug 14 2014

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..100

Cf. A242101 (by alternating group), A000041 (by symmetric group itself), A061417 (by cyclic group).

List([1..11],n->Size(OrbitsDomain(DihedralGroup(IsPermGroup,2*n),SymmetricGroup(IsPermGroup,n),\^)));

def a(n) : return (sum(euler_phi(n//d) * (n//d)^d * factorial(d) for d in divisors(n))//n + [(factorial(n//2) + factorial((n+1)//2 - 1)) * 2^(n//2-1), factorial((n-1)//2) * 2^((n-1)//2)][n%2]) // 2 # Eric M. Schmidt, Aug 23 2014

a(n) = (A061417(n) + b(n))/2, where b(n) = ((n-1)/2)! * 2^((n-1)/2) if n is odd, b(n) = ((n/2)! + (n/2-1)!) * 2^(n/2-1) if n is even. - Eric M. Schmidt, Aug 23 2014

More terms from Eric M. Schmidt, Aug 23 2014

A047917 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk!/n if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 2, 0, 6, 4, 0, 0, 0, 24, 2, 6, 8, 0, 0, 120, 6, 0, 0, 0, 0, 0, 720, 4, 8, 0, 48, 0, 0, 0, 5040, 6, 0, 36, 0, 0, 0, 0, 0, 40320, 4, 20, 0, 0, 384, 0, 0, 0, 0, 362880, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 4, 12, 64, 324, 0, 3840, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

			1; 1,1; 2,0,2; 2,2,0,6; 4,0,0,0,24; 2,6,8,0,0,120; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

Divide n-th row of A047916 by n.
Row sums give A061417.
Cf. A002024.

a047917 n k = a047917_tabl !! (n-1) !! (k-1)
a047917_row n = a047917_tabl !! (n-1)
a047917_tabl = zipWith (zipWith div) a047916_tabl a002024_tabl
-- Reinhard Zumkeller, Mar 19 2014

a[n_, k_] := If[ Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!/n, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]](* Jean-François Alcover, Feb 17 2012 *)

Offset corrected by Reinhard Zumkeller, Mar 19 2014

A242101 Number of conjugacy classes of the symmetric group S_n when conjugating only by even permutations.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 44, 58, 80, 104, 138, 180, 236, 302, 390, 496, 634, 800, 1010, 1264, 1586, 1970, 2448, 3024, 3734, 4582, 5622, 6862, 8372, 10168, 12336, 14912, 18010, 21672, 26052, 31226, 37384, 44632, 53226, 63318, 75238, 89202, 105630, 124832

Offset: 1

Attila Egri-Nagy, Aug 14 2014

nonn

Cf. A242099 (by dihedral group), A000041 (by symmetric group itself), A061417 (by cyclic group).

Cf. A046682.

List([1..11], n->Size(OrbitsDomain(AlternatingGroup(IsPermGroup, n), SymmetricGroup(IsPermGroup, n), \^)));

For n >=2, a(n) = A000041(n) + A000700(n) = 2*A046682(n) [by a formula in A046682]. - Eric M. Schmidt, Aug 23 2014

More terms from Eric M. Schmidt, Aug 23 2014

cp's OEIS Frontend

A064649 Row sums of the table A047916.

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A064636 Number of derangements up to cyclic rotations; permutation siteswap necklaces, with no fixed points (no "zero-throws", i.e., empty hands, if we use the mapping Perm2SiteSwap1 of A060495 and A060498).

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Maple

Mathematica

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A242099 Number of conjugacy classes of the symmetric group S_n when conjugating by the dihedral group D_n.

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GAP

Sage

Formula

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A047917 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k!/n if k|n else 0 (1<=k<=n).

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Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A242101 Number of conjugacy classes of the symmetric group S_n when conjugating only by even permutations.

Views

Author

Keywords

Crossrefs

Programs

GAP

Formula

Extensions

A047917 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk!/n if k|n else 0 (1<=k<=n).