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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. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

REFERENCES

Wilf, Herbert S. "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.

LINKS

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

FORMULA

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

EXAMPLE

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;

...

MAPLE

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

MATHEMATICA

<<Combinatorica`; Table[Distribution[Apply[LCM, Map[Length, Map[ToCycles, Permutations[n]], {2}], 1], Range[Max[Apply[LCM, IntegerPartitions[n], 1]]]], {n, 1, 8}] // Grid

(* Second program: *)

row[n_] := (orders = PermutationOrder /@ GroupElements[SymmetricGroup[n]]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* 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 *)

PROG

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

(PARI) 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

CROSSREFS

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.

Sequence in context: A298804 A155788 A108073 * A126074 A108916 A119421

Adjacent sequences: A057728 A057729 A057730 * A057732 A057733 A057734

KEYWORD

nonn,tabf,easy,look,nice

AUTHOR

Roger Cuculière, Oct 29 2000

EXTENSIONS

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

STATUS

approved

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Last modified December 3 07:15 EST 2022. Contains 358512 sequences. (Running on oeis4.)