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
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.
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
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.
KEYWORD
AUTHOR
Roger Cuculière, Oct 29 2000
EXTENSIONS
More terms from N. J. A. Sloane, Nov 01 2000
STATUS
approved