A322384 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 13, 4, 1, 67, 21, 7, 1, 411, 131, 46, 11, 1, 2911, 950, 341, 101, 16, 1, 23563, 7694, 2871, 932, 197, 22, 1, 213543, 70343, 26797, 9185, 2311, 351, 29, 1, 2149927, 709015, 275353, 98317, 27568, 5119, 583, 37, 1, 23759791, 7867174, 3090544, 1141614, 343909, 73639, 10366, 916, 46, 1

Offset: 1

Author

Alois P. Heinz, Dec 05 2018

Keywords

• nonn
• tabl

Examples

			The 6 permutations of {1,2,3} are:
  (1)     (2) (3)
  (1,2)   (3)
  (1,3)   (2)
  (2,3)   (1)
  (1,2,3)
  (1,3,2)
so there are 13 elements in the first cycles, 4 in the second cycles and only 1 in the third cycles.
Triangle T(n,k) begins:
       1;
       3,     1;
      13,     4,     1;
      67,    21,     7,    1;
     411,   131,    46,   11,    1;
    2911,   950,   341,  101,   16,   1;
   23563,  7694,  2871,  932,  197,  22,  1;
  213543, 70343, 26797, 9185, 2311, 351, 29, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
• Wikipedia, Permutation

Crossrefs

Columns k=1-10 give: A028418, A332851, A332852, A332853, A332854, A332855, A332856, A332857, A332858, A332859.

Row sums give A001563.

T(2n,n) gives A332928.

Cf. A185105, A322383.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]]*x^i, {i, 1, Length[l]}], Sum[Binomial[n-1, j-1]*b[n-j, Sort[Append[l, j]]]*(j-1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, {}], x] // Rest;
Array[T, 12] // Flatten  (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)