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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 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 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 <

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Last modified August 8 01:56 EDT 2024. Contains 375018 sequences. (Running on oeis4.)