A332852 -id:A332852 - cp's OEIS frontend

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

Alois P. Heinz, Dec 05 2018

			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;
  ...

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

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.

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);

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 *)

A350015 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 2, 5, 1, 17, 7, 74, 46, 394, 311, 15, 2484, 2241, 315, 18108, 17627, 4585, 149904, 152839, 57897, 2240, 1389456, 1460944, 705600, 72800, 14257440, 15326180, 8673060, 1660120, 160460640, 175421214, 110271546, 31600800, 1247400, 1965444480, 2177730270, 1469308698, 559402272, 55135080

Offset: 0

Steven Finch, Dec 08 2021

If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      5,      1;
[4]     17,      7;
[5]     74,     46;
[6]    394,    311,    15;
[7]   2484,   2241,   315;
[8]  18108,  17627,  4585;
[9] 149904, 152839, 57897, 2240;
...

Alois P. Heinz, Rows n = 0..140, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000774.
Row sums give A000142.
Cf. A126074, A145877, A332852, A349979, A349980, A350016, A350273, A350274.

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
      b(n-j, sort([l[], j])[2..4])*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(lprint(T(n)), n=0..14);  # Alois P. Heinz, Dec 11 2021

b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[Append[l, j]][[2 ;; 4]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..floor(n/3)} k * T(n,k) = A332852(n) for n >= 3. - Alois P. Heinz, Dec 12 2021

cp's OEIS Frontend

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.

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