A153761 -id:A153761 - 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

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

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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