A057740 -id:A057740 - 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

A048099 Number of degree-n even permutations of order exactly 2.

Original entry on oeis.org

0, 0, 0, 3, 15, 45, 105, 315, 1323, 5355, 18315, 63855, 272415, 1264263, 5409495, 22302735, 101343375, 507711375, 2495918223, 11798364735, 58074029055, 309240315615, 1670570920095, 8792390355903, 46886941456575, 264381946998975, 1533013006902975, 8785301059346175, 50439885753378303

Offset: 1

Vladeta Jovovic

Andrew Howroyd, Table of n, a(n) for n = 1..200
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).

Cf. A001189, A051695. A column of A057740.

Table[Sum[Binomial[n , 4 i] (4 i)!/(2^(2 i) (2 i)!), {i, 1, Floor[n/4]}], {n,1,22}] (* Luis Manuel Rivera Martínez, May 16 2018 *)

a(n) = sum(i=1, n\4, binomial(n,4*i)*(4*i)!/(2^(2*i)*(2*i)!)); \\ Michel Marcus, May 17 2018

seq(n)={my(A=O(x*x^n)); Vec(serlaplace(exp(x + x^2/2 + A) + exp(x - x^2/2 + A) - 2*exp(x + A))/2, -n)} \\ Andrew Howroyd, Feb 01 2020

a(n) = (A001189(n) + A051684(n))/2.
a(n) = Sum_{i=1..floor(n/4)} binomial(n,4i)(4i)!/(2^(2i)(2i)!). - Luis Manuel Rivera Martínez, May 16 2018
E.g.f.: (exp(x + x^2/2) + exp(x - x^2/2))/2 - exp(x). - Andrew Howroyd, Feb 01 2020

A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 7, 15, 15, 21, 21, 35, 35, 60, 105, 105, 105, 140, 210, 210, 420, 420, 420, 420, 840, 1155, 1260, 1365, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 15015, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060

Offset: 0

Vladeta Jovovic

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Index entries for sequences related to groups

Cf. A057742, A057743, A057740, A000793.

(* a3 = A000793  a4 = A051704 *) a3[n_] := Max[LCM @@@ IntegerPartitions[n]]; a4[n_] := (pp = Reap[ Do[ pk = p^k; If[pk <= n, Sow[pk]], {p, Prime[ Range[2, PrimePi[n]]]}, {k, 1, Ceiling[ Log[3, n]]}]][[2, 1]]; sel = Select[ IntegerPartitions[n, All, pp], Length[#] == Length[ Union[#] && !MatchQ[#, {_, x_, _, y_, _} /; GCD[x, y] != 1]] &]; Max[Times @@@ sel]); a4[0] = 1; a4[1] = a4[2] = a4[4] = a4[6] = 0; a[n_] := Max[a3[n - 2], a4[n - 1], a4[n]]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Sep 11 2012, from formula *)

a(n)={my(m=1); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, m=max(m, lcm(Vec(p))))); m} \\ Andrew Howroyd, Jul 03 2018

a(n)=max{ A000793(n-2), A051704(n-1), A051704(n) }, a(0)=a(1)=1.

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

A145437 a(n) = number of elements of order n in simple group Alt(12) of order 239500800.

Original entry on oeis.org

1, 63855, 776600, 3825360, 4809024, 25530120, 570240, 29937600, 26611200, 25945920, 43545600, 21621600, 0, 8553600, 6652800, 0, 0, 0, 0, 11975040, 11404800, 0, 0, 0, 0, 0, 0, 0, 0, 3991680, 0, 0, 0, 0, 13685760

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

t1:=[0 : n in [1..240]];
G:=Alt(12);
t2:=Classes(G);
for c in t2 do
t1[c[1]] := t1[c[1]] + c[2];
end for;
t1;

A145822 a(n) = number of elements of order n in simple group Alt(11) of order 19958400.

Original entry on oeis.org

1, 18315, 142010, 457380, 809424, 2044350, 237600, 2494800, 2217600, 498960, 3628800, 2910600, 0, 712800, 887040, 0, 0, 0, 0, 997920, 1900800

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A102578 a(n) = number of elements of order n in simple group Alt(6) = L_2(9) of order 360.

Original entry on oeis.org

1, 45, 80, 90, 144

Offset: 1

N. J. A. Sloane, Apr 06 2009

Cf. A031963. A row of A057740.

A145752 a(n) = number of elements of order n in simple group Alt(7) of order 2520.

Original entry on oeis.org

1, 105, 350, 630, 504, 210, 720

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A145753 a(n) = number of elements of order n in simple group Alt(8) of order 20160.

Original entry on oeis.org

1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A145754 a(n) = number of elements of order n in simple group Alt(9) of order 181440.

Original entry on oeis.org

1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

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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A048099 Number of degree-n even permutations of order exactly 2.

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A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

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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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A145437 a(n) = number of elements of order n in simple group Alt(12) of order 239500800.

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Magma

A145822 a(n) = number of elements of order n in simple group Alt(11) of order 19958400.

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A102578 a(n) = number of elements of order n in simple group Alt(6) = L_2(9) of order 360.

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A145752 a(n) = number of elements of order n in simple group Alt(7) of order 2520.

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A145753 a(n) = number of elements of order n in simple group Alt(8) of order 20160.

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A145754 a(n) = number of elements of order n in simple group Alt(9) of order 181440.

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