A060014 -id:A060014 - cp's OEIS frontend

A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 20, 30, 24, 20, 1, 75, 80, 180, 144, 240, 1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420, 1, 763, 1232, 5460, 1344, 10640, 5760, 5040, 0, 4032, 0, 3360, 0, 0, 2688, 1, 2619, 5768, 30996, 3024, 83160, 25920, 45360, 40320, 27216, 0, 30240, 0, 25920, 24192, 0, 0, 0, 0, 18144

Offset: 1

Roger Cuculière, Oct 29 2000

Every row for n >= 7 contains zeros. Landau's function quickly becomes > 2*n, and there is always a prime between n and 2*n. T(n,p) = 0 for such a prime p. - Franklin T. Adams-Watters, Oct 25 2011

			Triangle begins:
  1;
  1,   1;
  1,   3,   2;
  1,   9,   8,   6;
  1,  25,  20,  30,  24,   20;
  1,  75,  80, 180, 144,  240;
  1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420;
  ...

Herbert S. Wilf, "The asymptotics of e^P(z) and the number of elements of each order in S_n." Bull. Amer. Math. Soc., 15.2 (1986), 225-232.

Alois P. Heinz, Rows n = 1..30, flattened
FindStat - Combinatorial Statistic Finder, The order of a permutation.
Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages). - _N. J. A. Sloane_, Mar 27 2015

Cf. A000793, also A054522 (for cyclic group), A057740 (alternating group), A057741 (dihedral group).
Rows sums give A000142, last elements of rows give A074859, columns k=2, 3, 5, 7, 11 give A001189, A001471, A059593, A153760, A153761. - Alois P. Heinz, Feb 16 2013
Main diagonal gives A074351.
Cf. A222029.

Magma
```
{* Order(g) : g in Sym(6) *};
```

with(group):
for n from 1 do
    f := [seq(0,i=1..n!)] ;
    mknown := 0 ;
    # loop through the permutations of n
    Sn := combinat[permute](n) ;
    for per in Sn do
        # write this permutation in cycle notation
        gen := convert(per,disjcyc) ;
        # compute the list of lengths of the cycles, then the lcm of these
        cty := [seq(nops(op(i,gen)),i=1..nops(gen))] ;
        if cty <> [] then
            lcty := lcm(op(cty)) ;
        else
            lcty := 1 ;
        end if;
        f := subsop(lcty = op(lcty,f)+1,f) ;
        mknown := max(mknown,lcty) ;
    end do:
    ff := add(el,el=f) ;
    print(seq(f[i],i=1..mknown)) ;
end do: # R. J. Mathar, May 26 2014
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jul 11 2017

<Jean-François Alcover, Aug 31 2016 *)
b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j-1)!*b[n-j, LCM[g, j]]* Binomial[n-1, j-1], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 03 2019, after Alois P. Heinz *)

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

Sum_{k=1..A000793(n)} k*T(n,k) = A060014(n); A000793 = Landau's function.

More terms from N. J. A. Sloane, Nov 01 2000

A028418 Sum over all n! permutations of n letters of maximum cycle length.

Original entry on oeis.org

1, 3, 13, 67, 411, 2911, 23563, 213543, 2149927, 23759791, 286370151, 3734929903, 52455166063, 788704078527, 12648867695311, 215433088624351, 3884791172487903, 73919882720901823, 1480542628345939807, 31128584449987511871, 685635398619169059391

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

Sum the n-permutations having at least 1 cycle of length >= i for all i >= 1. A000142 + A033312 + A066052 + A202364 + ... The summation is precisely that indicated in the title since each permutation whose longest cycle = i is counted i times. - Geoffrey Critzer, Jan 09 2013

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 183.
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 358.

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 142 terms from Thomas Dybdahl Ahle)
Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions, p. 22.

Cf. A006128, A028417, A060014, A126074.

Column k=1 of A322384.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, max(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

kmax = 19; gf[x_] = Sum[ 1/(1-x) - 1/(E^((x^(1+k)*Hypergeometric2F1[1+k, 1, 2+k, x])/ (1+k))*(1-x)), {k, 0, kmax}];
a[n_] := n!*Coefficient[Series[gf[x], {x, 0, kmax}], x^n]; Array[a, kmax]
(* Jean-François Alcover, Jun 22 2011, after e.g.f. *)
a[ n_] := If[ n < 1, 0, 1 + Total @ Apply[ Max, Map[ Length, First /@ PermutationCycles /@ Drop[ Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)

E.g.f.: Sum_{k>=0} (1/(1-x) - exp(Sum_{j=1..k} x^j/j)).
a(n) = f(n, 0, n, n!) where f(L, r, n, m) = m*r if r >= l, otherwise Sum_{k=0..L-1} (f(k, max(L-k,r), n-1, m/n) + (n-L)*f(L, r, n-1, m/n)). - Thomas Dybdahl Ahle, Aug 15 2011
a(n) = Sum_{k=1..n} k * A126074(n,k). - Alois P. Heinz, May 17 2016

More terms from Vladeta Jovovic, Sep 19 2002

More terms from Thomas Dybdahl Ahle, Aug 15 2011

A060015 Sum of orders of all even permutations of n letters.

Original entry on oeis.org

1, 1, 7, 31, 211, 1411, 12601, 137047, 1516831, 18111751, 223179001, 2973194071, 46287964867, 835826439631, 15722804528341, 292673102609791, 5177400032329231, 102538737981192607, 2284570602107946601

Offset: 1

N. J. A. Sloane, Mar 17 2001

			For n = 4 there is 1 even permutation (1) of order 1, 3 even permutations (12)(34) etc. of order 2 and 8 (123) etc. of order 3, for a total of 31.

Joshua Harrington, Lenny Jones, and Alicia Lamarche, Characterizing Finite Groups Using the Sum of the Orders of the Elements, International Journal of Combinatorics, Volume 2014, Article ID 835125, 8 pages.

Cf. A060014.

g[list_]:=Total[list]!/Apply[Times,list]/Apply[Times,Table[Count[list,n]!,{n,1,20}]];f[list_]:=Apply[Plus,Table[Count[list,n],{n,2,20,2}]];Map[Total,Table[Map[g,Select[Partitions[n],EvenQ[f[#]]&]]*Map[Apply[LCM,#]&,Select[Partitions[n],EvenQ[f[#]]&]],{n,1,20}]]  (* Geoffrey Critzer, Mar 26 2013 *)

More terms from Vladeta Jovovic, Mar 18 2001

A290932 Sum of the LCM of cycle lengths over all endofunctions on [n].

Original entry on oeis.org

1, 1, 5, 40, 431, 5886, 96817, 1862890, 41043375, 1018584610, 28108489541, 853617865134, 28287119604955, 1015630741097350, 39273014068691145, 1627118268024495586, 71904849762914854703, 3375959341815207350850, 167810405947367539063885, 8803814897608815310714270

Offset: 0

Alois P. Heinz, Aug 13 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A000793, A060014, A208231, A208248, A222029.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

T[n_, k_] := T[n, k] = If[n == 0, k, Sum[(j - 1)! * T[n - j, LCM[k, j]]*Binomial[n - 1, j - 1], {j, n}]]; {1}~Join~Table[Sum[T[j, 1]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], {n, 19}] (* Michael De Vlieger, Aug 17 2017 *)

a(n) = Sum_{k=1..A000793(n)} k * A222029(n,k).

A060179 Sum of distinct orders of degree-n permutations.

Original entry on oeis.org

1, 1, 3, 6, 10, 21, 21, 50, 73, 116, 167, 248, 385, 496, 728, 959, 1548, 1899, 2835, 3609, 5042, 6403, 8336, 12187, 15522, 21358, 26090, 35298, 44147, 62512, 76289, 101403, 123883, 156880, 200086, 254175, 335380, 413184, 505860, 615258, 810767, 980747, 1293953

Offset: 0

Vladeta Jovovic, Mar 19 2001

			Set of orders of all degree 7 permutations is {1,2,3,4,5,6,7,10,12} so a(7)=1+2+3+4+5+6+7+10+12=50.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A060014, A060015.
Cf. A009490.
Row sums of A256553.

b:= proc(n, i) option remember; (p->`if`(i*n=0, 1,
       add(b(n-p^j, i-1)*p^j, j=1..ilog[p](n))+
         b(n, i-1)))(`if`(i=0, 0, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 12 2017

b[n_, i_] := b[n, i] = Function [p, If[i*n == 0, 1, Sum[b[n-p^j, i-1]*p^j, {j, 1, Floor@Log[p, n]}] + b[n, i-1]]][If[i == 0, 0, Prime[i]]];
a[n_] := b[n, PrimePi[n]];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

G.f.: Prod(p prime, 1 + Sum(k >= 1, p^k*x^(p^k))) / (1-x). - Vladeta Jovovic, Sep 18 2002

More terms from David Wasserman, May 29 2002

a(0)=1 prepended by Alois P. Heinz, Apr 01 2015

A346066 Sum of GCD of cycle lengths over all permutations of [n].

Original entry on oeis.org

0, 1, 3, 10, 45, 216, 1505, 9360, 84105, 730240, 7715169, 76204800, 1090114025, 11975040000, 185501455425, 2791872219136, 45361870178625, 690452066304000, 14415096609538625, 236887827111936000, 5448878874163974249, 108418310412206080000, 2381309423564793710625

Offset: 0

Alois P. Heinz, Jul 03 2021

nonn

			a(3) = 10 = 3+3+1+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3), (1)(2)(3).

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

Cf. A060014 (the same for LCM), A346085.

b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
      *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);

b[n_, g_] := b[n, g] = If[n == 0, g, Sum[(j - 1)!*
     b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A346085(n,k).

A143300 Decimal expansion of the Goh-Schmutz constant.

Original entry on oeis.org

1, 1, 1, 7, 8, 6, 4, 1, 5, 1, 1, 8, 9, 9, 4, 4, 9, 7, 3, 1, 4, 0, 4, 0, 9, 9, 6, 2, 0, 2, 6, 5, 6, 5, 4, 4, 4, 1, 6, 3, 1, 1, 5, 5, 1, 2, 0, 4, 1, 2, 8, 8, 4, 2, 6, 5, 0, 6, 2, 8, 6, 5, 1, 4, 0, 1, 6, 0, 5, 4, 5, 5, 1, 8, 4, 2, 0, 3, 8, 5, 9, 1, 8, 1, 4, 8, 8, 0, 0, 7, 3, 5, 6, 5, 0, 0, 5, 2, 7, 1, 2, 9, 1, 2, 7

Offset: 1

Eric W. Weisstein, Aug 05 2008

This constant is the limit of the logarithm expected order of a random permutation of length n, divided by sqrt(n/log n). In other words, log(A060014(n)/n!) ~ c sqrt(n/log n) where c is this constant. Stong improves the error term to O(sqrt(n) log log n/log n). - Charles R Greathouse IV, Nov 06 2014

			1.1178641511899449731...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 287.

P. Erdős and P. Turán, On some problems of a statistical group theory, IV, Acta Math. Acad. Sci. Hungar. 19 (1968), pp. 413-435. [alternate link]
William M. Y. Goh and Eric Schmutz, The expected order of a random permutation, Bulletin of the London Mathematical Society 23:1 (1991), pp. 34-42.
Richard Stong, The average order of a permutation, Electronic Journal of Combinatorics 5 (1998), 6 pp.
Eric Weisstein's World of Mathematics, Goh-Schmutz Constant

RealDigits[ N[ Integrate[Log[1 + t]/(E^t - 1), {t, 0, Infinity}], 105]][[1]] (* Jean-François Alcover, Oct 26 2012 *)

intnum(t=0,[oo,1],log(1+t)/(exp(t)-1)) \\ Charles R Greathouse IV, Nov 05 2014

From Amiram Eldar, Aug 13 2020: (Start)
Equals Integral_{x=0..oo} log(x+1)/(exp(x) - 1) dx.
Equals Integral_{x=0..oo} log(1 - log(1 - exp(-x))) dx.
Equals Integral_{x=0..oo} x*exp(-x)/((1 - exp(-x)) * (1 - log(1 - exp(-x)))) dx.
Equals -Sum_{k>=1} exp(k) * Ei(-k)/k, where Ei is the exponential integral. (End)

A382780 Sum of the orders of all permutations of [n] with distinct cycle lengths.

Original entry on oeis.org

1, 1, 2, 12, 48, 360, 2520, 22680, 221760, 2298240, 28425600, 385862400, 5269017600, 80951270400, 1347631084800, 21565729785600, 413922526617600, 8409043612569600, 172028224598630400, 3765253760710041600, 84080417596471296000, 1935910813364656128000

Offset: 0

Alois P. Heinz, May 11 2025

nonn

			a(3) = 12 = 2+2+2+3+3: (1)(23), (13)(2), (12)(3), (123), (132).

Alois P. Heinz, Table of n, a(n) for n = 0..170
Wikipedia, Permutation

Cf. A000142, A000793, A007838, A060014, A382781.

b:= proc(n, i, m) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..22);

A060178 Sum of orders of all odd permutations of n letters.

Original entry on oeis.org

0, 2, 6, 36, 260, 1860, 18732, 162176, 1774656, 20909160, 320781560, 4493532912, 72263548656, 1081552065776, 16682494491600, 315573151180800, 7042434106860032, 151228601744085216, 3306518316205792416

Offset: 1

Vladeta Jovovic, Mar 18 2001

nonn

Cf. A060014, A060015.

A067835 Average order of an element in the symmetric group S_n rounded down.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 10, 13, 15, 19, 21, 24, 29, 34, 39, 45, 51, 57, 66, 76, 86, 96, 108, 120, 134, 150, 170, 189, 208, 228, 253, 278, 310, 342, 375, 409, 449, 494, 544, 595, 645, 699, 761, 834, 909, 986, 1069, 1153, 1247, 1356, 1473, 1589, 1710, 1841

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), Feb 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A060014.

a(n) = floor(A060014(n) / n!).

More terms from Alois P. Heinz, Apr 01 2015

cp's OEIS Frontend

A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

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A028418 Sum over all n! permutations of n letters of maximum cycle length.

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A060015 Sum of orders of all even permutations of n letters.

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A290932 Sum of the LCM of cycle lengths over all endofunctions on [n].

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Mathematica

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A060179 Sum of distinct orders of degree-n permutations.

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Maple

Mathematica

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A346066 Sum of GCD of cycle lengths over all permutations of [n].

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Maple

Mathematica

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A143300 Decimal expansion of the Goh-Schmutz constant.

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