A322383 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 10, 7, 1, 45, 37, 13, 1, 236, 241, 101, 21, 1, 1505, 1661, 896, 226, 31, 1, 10914, 13301, 7967, 2612, 442, 43, 1, 90601, 117209, 78205, 29261, 6441, 785, 57, 1, 837304, 1150297, 827521, 346453, 88909, 14065, 1297, 73, 1, 8610129, 12314329, 9507454, 4338214, 1253104, 234646, 28006, 2026, 91, 1

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..141, flattened

Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.

Wikipedia, Permutation

Example

The 6 permutations of {1,2,3} are:
  (1)     (2)   (3)
  (1)     (2,3)
  (2)     (1,3)
  (3)     (1,2)
  (1,2,3)
  (1,3,2)
so there are 10 elements in the first cycles, 7 in the second cycles and only 1 in the third cycles.
Triangle T(n,k) begins:
      1;
      3,      1;
     10,      7,     1;
     45,     37,    13,     1;
    236,    241,   101,    21,    1;
   1505,   1661,   896,   226,   31,   1;
  10914,  13301,  7967,  2612,  442,  43,  1;
  90601, 117209, 78205, 29261, 6441, 785, 57, 1;
  ...

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, l.x^Range[Length[l]], Sum[Binomial[n - 1, j - 1] b[n - j, Sort[Append[l, j]]] (j - 1)!, {j, 1, n}]];
T[n_] := Rest @ CoefficientList[b[n, {}], x];
Array[T, 12] // Flatten (* Jean-François Alcover, Mar 03 2020, after Alois P. Heinz *)

Crossrefs

Columns k=1-10 give: A028417, A332906, A332907, A332908, A332909, A332910, A332911, A332912, A332913, A332914.

Row sums give A001563.

Cf. A185105, A322384.

Sequence in context: A117207 A046658 A124574 * A295856 A052964 A084178

Adjacent sequences: A322380 A322381 A322382 * A322384 A322385 A322386

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 05 2018

Status

approved