A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

1, 3, 1, 12, 5, 1, 60, 27, 8, 1, 360, 168, 59, 12, 1, 2520, 1200, 463, 119, 17, 1, 20160, 9720, 3978, 1177, 221, 23, 1, 181440, 88200, 37566, 12217, 2724, 382, 30, 1, 1814400, 887040, 388728, 135302, 34009, 5780, 622, 38, 1, 19958400, 9797760, 4385592, 1606446, 441383, 86029, 11378, 964, 47, 1

Offset: 1

Author

Wouter Meeussen, Dec 26 2012

Keywords

• nonn
• tabl

Comments

Row sums are n!*n = A001563(n) (see example).

For fixed k>=1, A185105(n,k) ~ n!*n/2^k. - Vaclav Kotesovec, Apr 25 2017

Examples

			The six permutations of n=3 in ordered cycle form are:
{ {1}, {2}, {3}    }
{ {1}, {2, 3}, {}  }
{ {1, 2}, {3}, {}  }
{ {1, 2, 3}, {}, {}}
{ {1, 3, 2}, {}, {}}
{ {1, 3}, {2}, {}  }
.
The lengths of the cycles in position k=1 sum to 12, those of the cycles in position k=2 sum to 5 and those of the cycles in position k=3 sum to 1.
Triangle begins:
       1;
       3,     1;
      12,     5,     1;
      60,    27,     8,     1;
     360,   168,    59,    12,    1;
    2520,  1200,   463,   119,   17,   1;
   20160,  9720,  3978,  1177,  221,  23,  1;
  181440, 88200, 37566, 12217, 2724, 382, 30, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Wikipedia, Permutation

Crossrefs

Columns k=1-10 give: A001710(n+1), A138772, A159324(n-1)/2 or A285231, A285232, A285233, A285234, A285235, A285236, A285237, A285238.

T(2n,n) gives A285239.

Cf. A000142, A001563, A270236, A284816, A285439.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add((p-> p+coeff(p, x, 0)*j*x^i)(b(n-j, i+1))*
       binomial(n-1, j-1)*(j-1)!, j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 15 2017

Mathematica

Table[it = Join[RotateRight /@ ToCycles[#], Table[{}, {k}]] & /@ Permutations[Range[n]]; Tr[Length[Part[#, k]]& /@ it], {n, 7}, {k, n}]
(* Second program: *)
b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 30 2018, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Apr 15 2017