A057731 -id:A057731 - cp's OEIS frontend

A074759 Number of degree-n permutations of order dividing n. Number of solutions to x^n = 1 in S_n.

1, 1, 2, 3, 16, 25, 396, 721, 11264, 46089, 602200, 3628801, 133494912, 479001601, 7692266960, 95904273375, 1914926104576, 20922789888001, 628693317946656, 6402373705728001, 182635841123840000, 2496321046987530021, 55826951075231672512, 1124000727777607680001

Offset: 0

Vladeta Jovovic, Sep 28 2002

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A057731, A074351, A261431.

Main diagonal of A008307.

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=numtheory[divisors](k))))
    end:
a:= n-> A(n, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

Table[a = Sum[x^i/i, {i, Divisors[n]}]; Part[Range[0, 20]! CoefficientList[Series[Exp[a], {x, 0, 20}], x],n + 1], {n, 0, 20}]  (* Geoffrey Critzer, Dec 04 2011 *)

a(n) = n! * [x^n] exp(Sum_{k divides n} x^k/k).

a(n) = Sum_{d|n} A057731(n,d) for n >= 1. - Alois P. Heinz, Jul 05 2021

A171806 Number of 5 X 5 permutation matrices such that the n-th matrix power is the least nonnegative power that gives the identity matrix.

Original entry on oeis.org

1, 25, 20, 30, 24, 20

Offset: 1

Artur Jasinski, Dec 18 2009

The sum of the terms of this sequence is equal to the number of 5 X 5 permutation matrices: 5! = 120.

Number of elements of order n in symmetric group S_5. - Alois P. Heinz, Mar 30 2020

			a(1) = 1 because there is only one matrix whose first power is the identity matrix (this is the identity matrix itself).

Row n=5 of A057731.

tab = {0, 0, 0, 0, 0, 0}; per =
 Permutations[{1, 2, 3, 4, 5}]; zeromat = {}; Do[
 AppendTo[zeromat, Table[0, {n, 1, 5}]], {m, 1, 5}]; unit =
 IdentityMatrix[5]; s5 = {}; Do[s = zeromat;
 Do[s[[m]][[per[[n]][[m]]]] = 1, {m, 1, 5}];
 AppendTo[s5, s], {n, 1, 120}]; Do[
 If[MatrixPower[s5[[n]], 1] == unit, tab[[1]] = tab[[1]] + 1,
  If[MatrixPower[s5[[n]], 2] == unit, tab[[2]] = tab[[2]] + 1,
   If[MatrixPower[s5[[n]], 3] == unit, tab[[3]] = tab[[3]] + 1,
    If[MatrixPower[s5[[n]], 4] == unit, tab[[4]] = tab[[4]] + 1,
     If[MatrixPower[s5[[n]], 5] == unit, tab[[5]] = tab[[5]] + 1,
      If[MatrixPower[s5[[n]], 6] == unit,
       tab[[6]] = tab[[6]] + 1]]]]]], {n, 1, 120}]; tab

Name edited and terms corrected by Alois P. Heinz, Mar 30 2020

A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 2, 0, 15, 3, 0, 6, 0, 96, 0, 0, 0, 24, 0, 455, 105, 40, 0, 0, 120, 0, 4320, 0, 0, 0, 0, 0, 720, 0, 29295, 4725, 0, 1260, 0, 0, 0, 5040, 0, 300160, 0, 22400, 0, 0, 0, 0, 0, 40320, 0, 2663199, 530145, 0, 0, 72576, 0, 0, 0, 0, 362880

Offset: 0

Alois P. Heinz, Jul 04 2021

			T(3,1) = 4: (1)(23), (13)(2), (12)(3), (1)(2)(3).
T(4,4) = 6: (1234), (1243), (1324), (1342), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,      1;
  0,       4,      0,     2;
  0,      15,      3,     0,    6;
  0,      96,      0,     0,    0,    24;
  0,     455,    105,    40,    0,     0, 120;
  0,    4320,      0,     0,    0,     0,   0, 720;
  0,   29295,   4725,     0, 1260,     0,   0,   0, 5040;
  0,  300160,      0, 22400,    0,     0,   0,   0,    0, 40320;
  0, 2663199, 530145,     0,    0, 72576,   0,   0,    0,     0, 362880;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-1 give: A000007, A079128.
Even bisection of column k=2 gives A346086.
Row sums give A000142.
T(2n,n) gives A110468(n-1) for n >= 1.
Cf. A057731, A145877, A346066, A359951.

b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);

b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*
     b[n - j, GCD[g, j]] Binomial[n - 1, j - 1], {j, n}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A346066(n).

Sum_{prime p <= n} T(n,p) = A359951(n). - Alois P. Heinz, Jan 20 2023

A074881 Triangle T(n,k) giving number of labeled cyclic subgroups of order k in symmetric group S_n, n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 4, 3, 1, 25, 10, 15, 6, 10, 1, 75, 40, 90, 36, 120, 1, 231, 175, 420, 126, 735, 120, 126, 105, 1, 763, 616, 2730, 336, 5320, 960, 1260, 1008, 840, 336, 1, 2619, 2884, 15498, 756, 41580, 4320, 11340, 6720, 6804, 7560, 4320, 3024, 2268

Offset: 1

Vladeta Jovovic, Sep 30 2002

A057731 contains zeros. This sequence contains only positive values of A057731(n,k)/A000010(k). - Alois P. Heinz, Feb 16 2013

			Triangle begins:
  1;
  1,   1;
  1,   3,   1;
  1,   9,   4,   3;
  1,  25,  10,  15,   6,  10;
  1,  75,  40,  90,  36, 120;
  1, 231, 175, 420, 126, 735, 120, 126, 105;
  ...

Alois P. Heinz, Rows n = 1..42, flattened

Row sums give A051625.

Cf. A000010, A181949.

nmax = 10;
T[n_, k_] := n! SeriesCoefficient[O[x]^(n+1) + Sum[MoebiusMu[k/i]*Exp[ Sum[x^j/j, {j, Divisors[i]}]], {i, Divisors[k]}], {x, 0, n}]/ EulerPhi[k];
Table[DeleteCases[Table[T[n, k], {k, 1, 2 nmax}], 0], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Sep 16 2019, after Andrew Howroyd *)

T(n,k)={n!*polcoeff(sumdiv(k, i, moebius(k/i)*exp(sumdiv(i, j, x^j/j) + O(x*x^n))), n)/eulerphi(k)} \\ Andrew Howroyd, Jul 02 2018

T(n,k) = A057731(n,k)/A000010(k).

A153760 Number of degree-n permutations of order exactly 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 5760, 25920, 86400, 237600, 570240, 1235520, 892045440, 13348249200, 106757164800, 604924594560, 2722120577280, 10344007402560, 34479959558400, 24928970490633600, 546446134633639680, 6281586217487489040, 50248618811434961280

Offset: 1

Franz Vrabec, Jan 01 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200

Column k=7 of A057731. - Alois P. Heinz, Feb 16 2013

a:= proc(n) option remember;
      `if`(n<7, 0, a(n-1)+(1+a(n-7))*(n-1)!/(n-7)!)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013

Rest[CoefficientList[Series[-Exp[x] + Exp[x + 1/7*x^7], {x, 0, 25}], x]*Range[0, 25]!] (* G. C. Greubel, Aug 27 2016 *)

seq(n)={Vec(serlaplace(-exp(x + O(x*x^n)) + exp(x + x^7/7 + O(x*x^n))),-n)} \\ Andrew Howroyd, Jul 02 2018

E.g.f.: -exp(x)+exp(x+1/7*x^7). - Alois P. Heinz, Feb 16 2013

More terms from Alois P. Heinz, Feb 16 2013

A214003 Number of degree-n permutations of prime order.

Original entry on oeis.org

0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109

Offset: 1

Stephen A. Silver, Feb 15 2013

nonn

			The symmetric group S_5 has 25 elements of order 2, 20 elements of order 3, and 24 elements of order 5. All other elements are of nonprime order (1, 4, or 6), so a(5) = 25 + 20 + 24 = 69.

Stephen A. Silver, Table of n, a(n) for n = 1..451

Cf. A001189, A001471, A057731, A059593, A153760, A153761, A186202.

b:= proc(n,p) option remember;
      `if`(n add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Jan 19 2023

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n],PrimeQ[Apply[LCM, #]] &]]], {n, 1,23}] (* Geoffrey Critzer, Nov 08 2015 *)

a(n) = Sum_{p prime} A057731(n,p).

E.g.f.: exp(x)*Sum_{p in Primes} exp(x^p/p)-1. - Geoffrey Critzer, Nov 08 2015

A153761 Number of degree-n permutations of order exactly 11.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 43545600, 283046400, 1320883200, 4953312000, 15850598400, 44910028800, 115482931200, 274271961600, 609493248000, 1279935820800, 4644633666390681600, 106826520356358566400, 1281918194457262387200, 10682651561168805120000

Offset: 1

Franz Vrabec, Jan 01 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200

Column k=11 of A057731. - Alois P. Heinz, Feb 16 2013

a:= proc(n) option remember;
      `if`(n<11, 0, a(n-1)+(1+a(n-11))*(n-1)!/(n-11)!)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013

Rest[CoefficientList[Series[-Exp[x] + Exp[x + 1/11*x^11], {x, 0, 25}], x]*Range[0, 25]!] (* G. C. Greubel, Aug 27 2016 *)

E.g.f.: -exp(x)+exp(x+1/11*x^11). - Alois P. Heinz, Feb 16 2013

More terms from Alois P. Heinz, Feb 16 2013

A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 2, 1, 5, 0, 2, 1, 5, 0, 0, 4, 1, 7, 2, 0, 0, 2, 1, 7, 0, 0, 0, 0, 6, 1, 9, 0, 2, 0, 0, 0, 4, 1, 9, 2, 0, 0, 0, 0, 0, 6, 1, 11, 0, 0, 4, 0, 0, 0, 0, 4, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 13, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 15, 0, 0, 0, 0, 6, 0, 0

Offset: 1

Roger Cuculière, Oct 29 2000

Note that D_2 equals the cyclic group of order 2.

			1,1;
1,3;
1,3,2;
1,5,0,2;
1,5,0,0,4; ...

Cf. A057731, A054522, A057740.

t[n_, k_] /; k != 2 && ! Divisible[n, k] = 0; t[n_, k_] /; k != 2 && Divisible[n, k] := EulerPhi[k]; t[n_, 2] := n + 1 - Mod[n, 2]; Flatten[Table[t[n, k], {n, 1, 14}, {k, 1, If[n == 1, 2, n]}]] (* Jean-François Alcover, Jun 19 2012, from formula *)
row[n_] := (orders = PermutationOrder /@ GroupElements[DihedralGroup[n]]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

If k<>2 and k does not divide n, this number is 0; if k<>2 and k divides n, this number is phi(k), where phi is the Euler totient function; if k=2, this number is n for odd n and n+1 for even n.

More terms from James Sellers, Oct 30 2000

A339122 Number of elements of the Rubik's Cube group of order A338883(n).

Original entry on oeis.org

1, 170911549183, 33894540622394, 4346957030144256, 133528172514624, 140621059298755526, 153245517148800, 294998638981939200, 55333752398428896, 34178553690432192, 44590694400, 2330232827455554048, 23298374383021440, 14385471333209856, 150731886270873600

Offset: 1

Ben Whitmore, Nov 24 2020

The most common order is 60, with a(33) = 4199961633799421952 elements, or about 9.71% of the group.

The least common order (excluding 1) is 11, with a(11) = 44590694400 elements, or about 0.0000001% of the group. Elements of order 11 are rare because they cannot affect the corner pieces of the cube.

			a(1) = 1 because the only element of order A338883(1) = 1 is the identity element.
a(73) = 51490480088678400 is the number of elements of order A338883(73) = 1260.

Ben Whitmore, Table of n, a(n) for n = 1..73
Tomas Rokicki, SpeedSolving Puzzles Community.

Cf. A057731, A075152, A338883.

pN[p_] := Total[p]!/Times@@p/Times@@Factorial[Flatten[Tally[p]][[2 ;; ;; 2]]]
oddQ[p_] := OddQ[Total[p - 1]]
ord[p_] := LCM @@ p
oriN[p_, o_] := Module[{i, t, a = 0, ns = 0, s = 0, r}, t = ord[p]/p;
  For[i = 1, i <= Length[p], i++,
   If[Mod[t[[i]], o] == 0, a += p[[i]], ns += 1; s += p[[i]]]];
     {If[a == 0, r = o^(s - ns), r = o^a o^(s - ns - 1)], o^(a + s - 1) - r}]
val[p1_, o1_, p2_, o2_] :=
Module[{z}, z = pN[p1] pN[p2];
     {{LCM[ord[p1], ord[p2]],z oriN[p1, o1][[1]] oriN[p2, o2][[1]]},
     {{LCM[ord[p1] o1,ord[p2]],z oriN[p1, o1][[2]] oriN[p2, o2][[1]]}},   {{LCM[ord[p1],ord[p2] o2],z oriN[p1, o1][[1]] oriN[p2, o2][[2]]}},
  {{LCM[ord[p1] o1, ord[p2] o2], z oriN[p1, o1][[2]] oriN[p2, o2][[2]] }}}]
p8 = IntegerPartitions[8]; p12 = IntegerPartitions[12];
ce = Select[p8, ! oddQ[#] &]; co = Select[p8, oddQ[#] &];
ee = Select[p12, ! oddQ[#] &]; eo = Select[p12, oddQ[#] &];
res = {}; max = 0;
For[i = 1, i <= Length[ce], i++,
For[j = 1, j <= Length[ee], j++,
  AppendTo[res, val[ce[[i]], 3, ee[[j]], 2]]]]
For[i = 1, i <= Length[co], i++,
For[j = 1, j <= Length[eo], j++,
  AppendTo[res, val[co[[i]], 3, eo[[j]], 2]]]]
p = Partition[res // Flatten, 2]; c // Clear;
For[i = 1, i <= Length[p], i++,
  If [IntegerQ[c[p[[i, 1]]]], c[p[[i, 1]]] += p[[i, 2]],
   c[p[[i, 1]]] = p[[i, 2]]]; If[p[[i, 1]] > max, max = p[[i, 1]]]];
Select[Table[c[i], {i, 1, max}], IntegerQ[#] &] (* Herbert Kociemba, Jun 30 2022 *)

# See post #11 in SpeedSolving Puzzles Community link.

Sum_{n=1..73} a(n) = 43252003274489856000 = A075152(3).

a(10) corrected by Ben Whitmore, Jun 27 2022

A345628 Irregular triangle T(n,k) read by rows of the number of elements of order k in the dicyclic group Dic(n) for n>=2.

Original entry on oeis.org

1, 1, 0, 6, 1, 1, 2, 6, 0, 2, 1, 1, 0, 10, 0, 0, 0, 4, 1, 1, 0, 10, 4, 0, 0, 0, 0, 4, 1, 1, 2, 14, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 0, 14, 0, 0, 6, 0, 0, 0, 0, 0, 0, 6, 1, 1, 0, 18, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 18, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0

Offset: 2

Sean A. Irvine, Jun 22 2021

Dic(1) is omitted since it is degenerate.

Row n has 2*n entries (k=1..2*n).

			Triangle begins:
  1, 1, 0,  6;
  1, 1, 2,  6, 0, 2;
  1, 1, 0,  1, 0, 0, 0, 4;
  1, 1, 0, 10, 4, 0, 0, 0, 0, 4;
  ...

Sean A. Irvine, Rows n=2..100 flattened
Sean A. Irvine, Java program (github)
Wikipedia, Dicyclic group

Cf. A054522, A057731.

cp's OEIS Frontend

A074759 Number of degree-n permutations of order dividing n. Number of solutions to x^n = 1 in S_n.

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A171806 Number of 5 X 5 permutation matrices such that the n-th matrix power is the least nonnegative power that gives the identity matrix.

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A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A074881 Triangle T(n,k) giving number of labeled cyclic subgroups of order k in symmetric group S_n, n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

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A153760 Number of degree-n permutations of order exactly 7.

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Maple

Mathematica

PARI

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A214003 Number of degree-n permutations of prime order.

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Maple

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A153761 Number of degree-n permutations of order exactly 11.

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Maple

Mathematica

Formula

Extensions

A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.

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